Datasets:

qingy2024

/

qwark-corpus

like 1

# Add single digit numbers with decimal values in thousandths Lesson When we add numbers, place value is important. When we work through a problem such as $456+349$456+349, we have to start at the right hand side, and work to the left. Adding whole numbers with three digits is an ideal way to move to adding decimals to three decimal places.  You may like to refresh your memory of adding numbers with one decimal place, before tackling numbers with three decimal places. ## With decimals in our numbers Having decimals in our numbers doesn't require us to do anything different, it's just that the value of our digits changes. Imagine if, now, we have $0.456+0.349$0.456+0.349. While the digits are the same, the value of the numbers are different. Here, instead of adding $456$456 to $349$349, we are adding $456$456 hundredths to $349$349 hundredths. Our answer is $805$805 hundredths, which is less than $1$1. So, while the digits are the same, the answer is $1000$1000 times smaller! In our first video, we look at adding numbers with decimals to three places. Remember! When we add decimals, we always start with the smallest value, which is the number furthest to the right. ## Renaming In the second video, you can see an example of adding two numbers with decimals but this time we need to regroup, or rename, some decimals. We use the same process that we use for whole numbers. Once again, we just have more steps to work through. ## Adding decimals with no place-value table Finally, in our third video, we work through an example without the place value table! Hopefully, you have seen how it helps and now feel confident to try this yourself. It's the same process, but we need to be careful with how we write our answers so that our numbers line up in the correct places. Oh, and we will also look at how to regroup from decimals to whole numbers. You'll soon see it's the same process we always use, we just have to keep moving left. Remember! It's also important to line up the place value columns. Estimating our answer is a great way to check that our decimal point is in the correct spot. #### Worked examples ##### Question 1 Find $1.007+0.1$1.007+0.1 giving your answer as a decimal. ##### Question 2 Find $3.003+2.007$3.003+2.007 giving your answer as a decimal. ##### Question 3 Find $2.066+6.025$2.066+6.025 giving your answer as a decimal.

crawl-data/CC-MAIN-2024-18/segments/1712296818105.48/warc/CC-MAIN-20240422082202-20240422112202-00533.warc.gz

The demography of the Federal Republic of Germany is monitored by the "Statistisches Bundesamt" (Federal Statistical Office of Germany). The population of Germany is approximately 82,220,000, making it the 14th most populous country in the world. Germany's population is characterized by zero or declining growth , with an aging population and smaller cohort of youths. More than 16 million people are of non-German descent (first and second generation, including mixed heritage), about seven million of which are foreign residents. The largest ethnic group of non-German origin are the Turkish. Since the 1960s, West and later reunified Germany has been attracting migrants primarily from Southern and Eastern Europe as well as Turkey, many of which (or their children) over time acquired German citizenship. While most of these migrations had an economic background, Germany has also been a prime destination for refugees from many developing countries, in part because its constitution long had a clause giving a 'right' to political asylum, but restrictions over the years have since made it less attractive. Within Germany, there is a long history of East-to-West migrations, starting with the 19th century Ostflucht. After the World War II border shifts and expulsions, the Germans from Eastern Europe and the former eastern territories moved westward to post-war Germany. During the partition of Germany, many Germans from East Germany fled to West Germany for political and also economic reasons. Since Germany's reunification, there are ongoing migrations from the eastern New Länder to the western Old Länder for economic reasons. Germany has one of the world's highest levels of education, technological development, and economic productivity. Since the end of World War II, the number of students entering universities has more than tripled, and the trade and technical schools are among the world's best. With a per capita income of about $27,000, Germany is a broadly middle class society. Germans also are very mobile; millions travel abroad each year. A generous social welfare system provides for universal health care, unemployment compensation, and other social needs. Due to Germany's aging population and struggling economy, the welfare system came under a lot of strain from the 1990s. This led the government to push through a wide-ranging programme of belt-tightening reforms, Agenda 2010, including the labour market reforms known as Hartz I - IV. In 2005 Germany had 82 cities with more than 100.000 inhabitants. |City name||Location||Description||Population (2004)||Largest German ethnic groups||Largest non-German ethnic groups| |Rhine-Ruhr Metropolitan Region||Cologne is the largest and unofficial capital city of the Rhineland, the very Western part of Germany. Particularly among young Germans, Cologne is renowned for its nightlife and open-minded atmosphere.||10.2 million||Rhinelanders, Westfalians and others||Turks, Poles, Italians, Dutch, French, Arabs, Iranians, South Asians like Indians, and Japanese (large Japanese community in Düsseldorf).| |Frankfurt Rhine-Main Region||Frankfurt is the economic and financial center both for Germany and the continental European Union. Frankfurt is arguably Germany's most international city. It boasts a large airport and numerous skyscrapers. Within Germany, the city has a reputation of being very business-oriented, perhaps at the expense of other pursuits.||5.8 million||Hessians and others||Turks, Italians, Arabs, Iranians, Greeks, Russians, Poles, Israelis, Koreans, and Pakistanis (mostly Pashtun & Panjabi's ethnic groups).| |Munich Metropolitan Region||Munich has Germany's highest standard of living. Countless sporting and leisure opportunities - both in the city and in its picturesque region. Munich is a powerhouse of the German economy and rich in Bavarian culture.||4.7 million||Bavarians, Franconians and others||Turks, Croats, Serbs, Greeks, Albanians, Macedonians, Italians, Bosnians, Hungarians, Spaniards and Romanians.| |Berlin/Brandenburg Metropolitan Region||Berlin is the capital of Germany and its largest city. Berlin lies in the eastern part of the country and has a reputation for cosmopolitan lifestyle, the German "city that never sleeps".||4.3 million||Berliners, Brandenburgians and others||Turks, Arabs, Poles, Russians, Albanians, Serbs,Macedonians, Vietnamese, Chinese, rising number of Africans, Chileans and other Latin Americans.| |Hamburg Metropolitan Region||Hamburg has a long tradition for sea trade and civil establishment. Hamburg is proud of its sophisticated bar and music scene and its reputation as Germany's "capital of good taste".||4.3 million||Lower Saxons and others||Turks, Russians, Albanians, Poles, Pakistanis, Iranians, Macedonians, Portuguese, Czechs and Slovaks.| |Leipzig-Halle-Dresden (Saxon Triangle)||Also dubbed "City of Heroes", Leipzig is where the 1989 revolution that brought down the Berlin Wall started. Today totally refurbished, it sports Europe's highest density of Art Nouveau architecture. Very lively bar scene, fastest growing economy in Germany.||3.5 million||Saxons and others||Vietnamese, Poles, Russians, Portuguese, Italians, Iranians, Turks, Arabs, Indians and Pakistanis.| |Stuttgart Metropolitan Region||Stuttgart has a reputation for research, inventions and industry. The German headquarters of many international enterprises are in Stuttgart. This contrasts with the strong rural, down-to-earth attitude of the Stuttgarters throughout the classes. A popular slogan is "We are good at everything. Except speaking High (standard) German."||3.5 million||Swabians and others||Turks, Greeks, Italians, Croats, Serbs, French, Chinese, Romanians, Americans and Spaniards.| |Bremen/Oldenburg Metropolitan Region||-||2.4 million||Lower Saxons, Frisians and others||Turks, Russians, Poles, Albanians, Serbs, Portuguese, Iranians, Dutch, Americans and Britons.| Four other sizable groups of people are referred to as "national minorities" (nationale Minderheiten) because they have lived in their respective regions for centuries: Danes, Frisians, Roma and Sinti, and Sorbs. Eastern and Northern Frisians live at Schleswig-Holstein's western coast, and in the north-western part of Lower Saxony. They are part of a wider community (Frisia) stretching from Germany to the northern Netherlands. The Sorbs, a Slavic people with about 60,000 members (according to government sources), are located in the Lusatia region of Saxony and Brandenburg. They are the last remnants of the Slavs that lived in central and eastern Germany since the 7th century. Roma people have been in Germany since the Middle Ages. They were persecuted by the Nazis, and thousands of Roma living in Germany were killed by the Nazi regime. Nowadays, they are spread all over Germany, mostly living in major cities. It is difficult to estimate their exact number, as the German government normally does not keep information on the ethnicity of its citizens. There are also many assimilated Sinti and Roma. A vague figure given by the German Department of the Interior is about 70,000. In the 1990s, many Roma moved to Germany from former Yugoslavia. In contrast to the old-established Roma population, the majority of them do not have German citizenship, they are classified as immigrants or refugees. After World War II, there was an influx of 14 million ethnic Germans from Eastern Europe who were expelled or fled after Germany lost the war. Since the 1960s, ethnic Germans from the Soviet Union have come to Germany, especially from Kazakhstan, Russia, and Ukraine. During the time of Perestroika, and after the dissolution of the Soviet Union, the number of immigrants increased heavily. Some of these immigrants are of mixed heritage. Germany now has Europe's third-largest Jewish population. In 2004, twice as many Jews from former Soviet republics settled in Germany as in Israel, bringing the total inflow to more than 200,000 since 1991. Jews have a voice in German public life through the Zentralrat der Juden in Deutschland. Some Jews from the former Soviet Union are of mixed heritage. Ethnic Diversity Compared Drastic changes in the socioeconomic landscape brought about by reunification have resulted in troubling social problems. Economic uncertainty in eastern Germany is often cited as one factor contributing to extremist violence, primarily from the political right. Confusion about the causes of the current hardships and a need to place blame have found expression in harassment and violence by some Germans directed toward foreigners, particularly non-Europeans. In its State of World Population 2006 report, the United Nations Population Fund lists Germany with hosting the third-highest percentage of the main international migrants worldwide, about 5% or 10 million of all 191 million migrants . Danish, Low German, the Sorbian languages (Lower Sorbian and Upper Sorbian), and the two Frisian languages, Saterfrisian and North Frisian, are officially recognized and protected as minority languages by the European Charter for Regional or Minority Languages in their respective regions. With speakers of Romany are living in all parts of Germany, the federal government has promised to take action to protect the language. Until now, only Hesse has followed Berlin's announcement, and agreed on implementing concrete measures to support Romany speakers. Implementation of the Charter is poor. The monitoring reports on charter implementation in Germany show many provisions unfulfilled. |Saterland Frisian||Lower Saxony| |Low German||Brandenburg, Bremen, Hamburg, Mecklenburg-Vorpommern, Lower Saxony, Saxony-Anhalt, Schleswig-Holstein, North Rhine-Westphalia| |Romany||Hesse (see text)| German dialects — some quite distinct from the standard language — are used in everyday speech, especially in rural regions. Many dialects, for example the Upper German varieties, are to some degree cultivated as symbols of regional identity and have their own literature, theaters and some few TV programming. While someone speaking dialect outside his home area might be frowned upon, in their original area some dialects can be spoken throughout all social classes. Nevertheless, partly due to Standard German media prevalence, their use has declined over the past century, especially in the younger population. The status of different German dialects can be very different. The Alemannic and Bavarian dialects of the south are positively valued by the speakers and can be used in almost all social circumstances. The Saxonian and Thuringian dialects have less prestige and are subject to derision. While Bavarian and Alemannic have kept much of their distinctiveness, the Middle German dialects, which are closer to Standard German, lost some of their distinctive lexical and grammatical features and tend to be only pronunciation variants of Standard German. Low Saxon is officially recognized as a language on its own, but despite this fact, there's little official action taken on fostering the language. Historically one third of Germany's territory and population was Low Saxon speaking. No data was ever collected on the actual number of speakers, but today the number of speakers ranges around 5 million persons. Despite this relatively high number of speakers there is very few coverage in the media (mostly on NDR TV, no regular programming) and very few education in or on the language. The language is not fixed as part of the school curriculum and Low Saxon is used as a medium of instruction in one school only in the whole Germany (as a "model project" in primary school sided by education in Standard German). As a consequence the younger generation refused to adopt the native language of their parents. Language prevalence dropped from more than 90% (depending on the exact region) in the 1930s to less than 5% today. This accounts for a massive intergenerational gap in language use. Older people regularly use the language and take private initiative to maintain the language, but the lack of innovative potential of the younger generation hinders language maintenance. The language too has an own literature (around 150 published books every year) and there are many theatres (mostly lay stages, but some professional ones, like for example Ohnsorg-Theater). Use of Low Saxon is mainly restricted to use under acquaintances, like family members, neighbours and friends. A meeting of a village council can be held almost completely in Low Saxon if all participants know each other (as long as written protocols are written in Standard German), but a single foreigner can make the whole switching to Standard German. The Low Saxon dialects are different in their status too. There's a north-south gradient in language maintenance. The Southern dialects of Westfalian, Eastfalian and Brandenburgish have had much stronger speaker losses, than the northern coastal dialects of Northern Low Saxon. While Eastfalian has lost speakers to Standard German, Westfalian has lost speakers to Standard German and Standard German based regiolect of the Rhine-Ruhr area. Brandenburgish speakers mostly switched to the Standard German based regiolect of Berlin. Brandenburgish is almost completely replaced by the Berlin regiolect. Northern Low Saxon speakers switched mostly to pure Standard German. English is the most common foreign language and almost universally taught by the secondary level. Other languages taught are French, Italian, Spanish, Portuguese, and Russian. Dutch is taught in counties bordering the Netherlands and Polish in the case of the eastern provinces facing Poland. Latin and Greek are part of the classical education syllabus offered in many secondary schools. According to a 2004 survey, two-thirds of Germany's citizens have at least basic knowledge of English. About 20% consider themselves to be speakers of French, followed by speakers of Russian (7%), Italian (6.1%), and Spanish (5.6%). The relatively high number of Russian speakers is a result of massive immigration from the former Soviet Union to Germany for almost 10 consecutive years — more than half of the Germans in the East learnt Russian at school, but only few people can speak it. Population growth rate: -0.12% (2007 est.) Birth rate: 8.3 births/1,000 population (2007 est.) Death rate: 10.1 deaths/1,000 population (2007 est.) Net migration rate: 0.49 migrant(s)/1,000 population (2007 est.) |at birth:||1.06 male(s)/female| |under 15 years:||1.05 male(s)/female| |15-64 years:||1.03 male(s)/female| |65 years and over:||0.62 male(s)/female| |total population:||0.96 male(s)/female (2000 est.)| Infant mortality rate: 4.09 deaths (within one year) per 1,000 live births (2007) Life expectancy at birth (2007): |total population:||78.6 years| Total fertility rate: 1.45 children born/woman (2007) In 1996, TFR was 1.39 for ethnic Germans and 2.40 for ethnic Turks.

<urn:uuid:01633d0f-37a4-498c-981b-10c88ce841ca>

Learning to read is a challenge for many students. According to the US Department of Education 1/3 of students cannot read at a “basic” level and only 1/3 achieve proficiency. And a 2020 Gallup analysis of U.S. Department of Education data found that more than half of Americans ages 16-74 read below a sixth-grade level. How come learning to read is such a challenge and what should we do about it? You’ve probably seen references to the “Science of Reading” in news stories about how America’s kids are failing in reading because they are not being given systematic and explicit instruction in “phonics.” This idea is best summed up by a quote from journalist Emily Hanford, one of the loudest voices on this subject. She wrote: “According to all the research, what you should see in every school is a heavy emphasis on explicit phonics instruction in the early grades.” At this point, the term “science of reading” has become synonymous with a heavy emphasis on phonics instruction. What the media means by “phonics” is a set of skills that teach students how letters combine to form sounds that represent the words of our language. Since written English is an alphabetic, phonetic language this makes sense. Kids do need to learn this. So “explicit phonics instruction” should be all kids need, right? Unfortunately, there are significant issues with this approach. The biggest problem is that English, though phonetic, is more irregular than any other Latin based language. For example, Italian is nearly perfectly phonetic. And that’s why children in Italy learn to decode their language in only 3-6 months. As neuroscientist Stanislas Dehaene notes: “One may wonder why English sticks to such a complicated spelling system. Indeed, Italians do not meet with the same problems. Their spelling is transparent: every letter maps onto a single phoneme, with virtually no exceptions. As a result, it only takes a few months to learn to read. This gives Italians an enormous advantage: their children’s reading skills surpass ours by several years, and they do not need to spend hours of schooling a week on dictation and spelling out loud. Furthermore, dyslexia is a much less serious problem for them.” Dehaene’s observation should not be surprising. English spelling has been studied by many researchers who have recorded just how irregular it is. For example, Dr. Godfrey Dewey, professor of orthography, conducted a detailed study of English used in books for elementary school students and found that only 1 in 5 words is spelled phonetically. And a landmark study conducted by Professor Paul Hanna funded by the US Dept. of Health, Education & Welfare found that 203 rules were required for a computer to obtain 50% accuracy in spelling. ghoti = fish? English spelling is so irregular that dreaming up hypothetical spellings for words has become a kind of pastime. For example, Professors Dorothea Simon and Herbert Simon found that the word “she” could be spelled 36 different ways (the “sh” sound can be spelled 9 ways – TI, SH, CI, SSI, SI, C, CH, T, S – and the “e” sound in 4 ways E, EA, EE, IE). Professor Julius Nyikos of Washington and Jefferson College found 1,768 ways of spelling 40 English phonemes – an average of 44 per sound. For instance, the “u” in the word “nut” can be spelled 60 different ways. And in what is probably the most famous example of orthographic gymnastics, in 1855 publisher Charles Ollier spelled the word fish as “ghoti” – taking gh from rough, o from women and ti from nation. (This example is frequently misattributed to George Bernard Shaw.) But while these examples are amusing, just consider how confusing it must be for a child first learning the written language. Professor Nyikos summed up the issue by writing, “It would be both ludicrous and tragic if it took lawsuits to jolt us into the realization that neither the teachers, nor the schools should be faulted as much as our orthography, which is incomparably more intricate than that of any other language. If English is not the absolute worst alphabetic spelling in the world, it is certainly among the most illogical, inconsistent, and confusing.” To get an idea of how complicated learning to “sound out” English is for a beginning reader, consider this one sentence where the very common “ea” vowel combination can be pronounced 13 different ways. That English spelling is highly irregular is simply a fact teachers and students have to deal with. That’s why, throughout history, a veritable who’s who of English speakers have promoted the idea of reforming English spelling. These include Benjamin Franklin, Samuel Johnson, Noah Webster, Andrew Carnegie, Charles Dickens, John Milton, George Bernard Shaw, Mark Twain, H.G. Wells, Upton Sinclair, Theodore Roosevelt, Charles Darwin, and Isaac Asimov, etc. But the spelling reformers have yet to succeed, and to this day English spelling is incredibly challenging for children (and adults). It’s because English spelling is so difficult to master that English is the ONLY language that has “spelling bee” competitions and a pronunciation guide in the dictionary. But many of the proponents of the “Science of Reading” state that teaching students phonics is a proven method to solve the problem. As Emily Hanford, stated emphatically “One of the most consistent findings in all of education research is that children become better readers when they get explicit and systematic phonics instruction.” But is this really true? The answer is, sadly, not really. What the research has shown consistently, is that when you give students explicit and systematic instruction in phonics, their scores go up on tests of phonics abilities. But the “science” does not show that students’ scores significantly increase on tests designed to assess students’ ability to read – that is fluently decode and comprehend text – especially over time. This is why Professor Jeffrey S. Bowers, after conducting an exhaustive and systematic review of 12 meta-analyses that assessed the efficacy of phonics, observed: “Systematic phonics did provide a moderate short-term benefit to regular word and pseudoword naming.” But, he concluded: “There is little or no empirical evidence that systematic phonics leads to better reading outcomes … There can be few areas in psychology in which the research community so consistently reaches a conclusion that is so at odds with available evidence.” (If you are interested in this subject and have not read this study, I urge you to. You can find the link to this research paper and every other paper cited in this article below.) So while kids do need to learn how our written language represents our spoken language, the use of explicit phonics instruction to teach them to read with fluency and comprehension has limited success – hence the perennial stories of a literacy “crisis.” What about all the “miracle” stories Science of Reading proponents like to cite? The simple answer is that they don’t hold up to scrutiny. Typically, these supposed gains result from: - small improvements resulting from significant but temporary changes in policy such as short-lived increases in education funding; - anchoring bias where the results only appear impressive because the students started so far behind; and most frequently, - harmful retention policies that keep the poorest students from appearing in the results. As Kevin Drum, who analyzed the reported Mississippi “miracle” concluded: “In 2013, Mississippi fourth graders are 13 points below the national average if you look at all students. After that year, “all students” excludes all the bottom kids who were held back. What we need to know is what the Mississippi scores would look like if we added them back in. And it turns out that Mississippi scores in 2022 are still about 13 points below the national average. In other words, the 2013 reforms had all but no effect.” Another problem with the “Science of Reading” is that the phonics texts they require give students a stilted, unrepresentative written language that is completely unlike the kind of text they encounter in real life, and as a result, are less effective at teaching students to read. As Ruth Price-Mohr and Colin Price noted: “Our results demonstrate a statistically significant difference and large effect size for reading comprehension in favor of low phonically-decodable texts. The findings challenge the assumption that children find highly decodable text easier to read.” “Science of Reading” proponents argue that students are not being taught explicit phonics and this is the problem. But the reality is that kids are being taught phonics and, because of the limitations built into the written English language, it’s just not as effective as we would hope. And the reason teachers are integrating other approaches such as “Balanced Literacy” or “Whole Language” is because using explicit phonics is not working for many of their students. So what should we do? We need to take a broader look at the “Science of reading” and incorporate the spectrum of skills that reading research shows are effective. As Professor Timothy Shanahan noted: “Any real “science of reading” would include all the methods or approaches that have been found, through research, to give kids a learning advantage in reading.” “Connected phonation is more effective than segmented phonation.” – Professor Linnea Ehri So we know that it’s essential for kids to learn how letters combine to represent the words of our spoken language. And we also know that English is highly irregular which makes it difficult to reliably sound out. So how do we resolve this? Since there are only two one-letter words in English (“a” and “I”) letter sounds are always combined. That process requires a skill called “blending” which can be challenging for new readers – especially since English is so irregular. Blending is the ability to combine individual sounds or phonemes to form words and the process of blending parts of words, often referred to as morphological awareness, is an important aspect of reading development. Moreover, blending morphemic units, such as prefixes, suffixes, and roots, can help early readers to derive meaning from unfamiliar words. By mastering blending skills, learners enhance their ability to decode unfamiliar words, improve reading fluency, and foster overall reading success. Research has shown that phonics blending supports students’ ability to read unfamiliar words because it provides them with a consistent strategy for approaching new words, and blending morphemic units, such as prefixes, suffixes, and roots, helps students derive meaning from unfamiliar words. As Professor Linnea Ehri concluded, “Connected phonation is more effective than segmented phonation for teaching beginning readers to decode unfamiliar words.” This is why Reading Kingdom has developed an innovative approach to teaching sound blending called “bit blends.” Instead of having a student do the complex but usual sound blend (e.g., “buh” + “ay” + “buh” + “ee” to make “baby”) using the bit blending process the program provides the initial or ending blend (e.g., “babe” with the word “baby”) and has the student add only a single sound (e.g., “ee”) to create the word. It’s a way to scaffold learning the blending process. As students’ exposures to letter and word sounds expand, their blending skills flourish. Additionally, Reading Kingdom supports students’ phonics learning by: - Reading a variety of texts (both narrative and expository) aloud to students as the words are highlighted on the screen. - Presenting words with their variations (such as eat, eats and eating) so students learn how words’ sounds change with prefixes and suffixes. - Allowing students to click on any word they see to hear what it “says.” This prevents guessing and the development of neural pathways for incorrect responses. - Using writing to reinforce phonemic awareness (more on this below). High Frequency Words “Beginning readers need to master a high-frequency vocabulary.” – Professor Edward Fry English has more words than any other language – over a million and growing. But surprisingly, about 120 of those words constitute 50%-60% of every page of text you will ever read. These are called “high frequency words” and leveraging their ubiquity gives kids a major leg up in both decoding and comprehension. The most common high frequency words are “function words” – articles (e.g., “a,” “an,” “the”), prepositions (e.g., “in,” “on,” “at”), conjunctions (e.g., “and,” “but,” “or”), and pronouns (e.g., “he,” “she,” “it”) – that are essential to English syntax. Professor Edward Fry, who was a Professor of Education and the Director of the Reading Center at Rutgers University and who is best known for creating the Fry Instant Words list, stated: “Beginning readers need to master a high-frequency vocabulary. They should be able to read these words “instantly” – without a moment’s hesitation – because these words make up 65 percent of all written material. In fact, more than half of the text of every newspaper article, textbook, children’s story, and novel is composed of these words. It is virtually impossible for students to concentrate on comprehension if they are stuck on a word such as ‘their.’” Fry’s and others’ research showed that knowing a sufficient number of high-frequency words helps students read more fluently and comprehend text more effectively. Other studies have shown that explicit instruction and practice with high-frequency words significantly improved students’ decoding skills, reading fluency, and comprehension. And knowing words by sight has additional benefits. As Professors Charles Hulme & Margaret J. Snowling observed, “When a reader’s eyes land on a familiar written word, its pronunciation, meaning, and syntactic role are all activated in memory.” Beyond their ubiquity, understanding the role function words play in English is crucial for reading comprehension. This is because these words are used to: |the boy, some toys, etc. |is here, was here, will be here, etc. |is running, are playing, etc. |What is? Did she? Are they? Etc. |Identify singular & plural |a girl, his home, they ran, these boxes, etc. |but, not, etc. |in the corner, on the box, etc. This is why Reading Kingdom explicitly teaches these 120 high frequency words – including their sounds, spellings, meanings and usages in context – and then leverages their power to help teach students the structure of English. This way, Reading Kingdom teaches students to both decode approximately 60% of every page they will ever read and to understand the relationships among all the other words on that page. This is an incredibly powerful method. “The evidence is clear: writing can be a vehicle for improving reading.” – Professors Steve Graham and Michael Hebert For decades researchers have emphasized the strong connection between reading and writing. Myriad studies have demonstrated that writing is a proven approach for improving reading. By engaging in writing activities, learners develop a deeper understanding of language, phonics, vocabulary, and overall literacy proficiency. Spelling is one key aspect of writing that is very beneficial for students. As Professor Rebecca Treiman noted, “learning to spell, in addition to its effects on spelling, also benefits children’s reading. The benefits of writing are both motivational and cognitive. In addition, spelling appears to have cognitive benefits. It encourages children to analyze words into smaller units of sound and to link these sounds to letters. In this way, children practice their phonemic segmentation skills. Through writing, children learn to see spellings as maps of phonemic content rather than as arbitrary sequences of letters … The research shows that spelling helps children master the alphabetic principle and also has a positive impact on reading.” In addition to spelling, writing also requires the ability to create connected text. Professors Steve Graham and Michael Hebert who conducted a meta-analysis of the impact of writing on reading found that “Teaching students how to write improves their reading comprehension, reading fluency, and word reading; and increasing how much students write enhances their reading comprehension.”24 And in a report they wrote which was published by the Alliance for Excellent Education, the authors concluded “writing instruction had a strong and consistent impact on improving students’ reading fluency.” 25 It is for these reasons that Reading Kingdom explicitly teaches students how to spell each word that is learned in the program through a variety of techniques and repeated exposures designed to stimulate students’ visual mapping and memory and to eliminate guessing. Then the program teaches students to use these words to write syntactically and semantically correct text starting with very simple texts at the beginning level and leading up to multi-sentence texts and story summaries at a Lexile level of 750 by the end of the program. “Any acquired knowledge must be stored in memory until it is used. Indeed, all learning depends on the ability of human memory to store such knowledge.” – Professor Frederick Reif While a majority of teaching focus in early literacy education is on sounds, it’s important to keep in mind that reading and writing are visual skills and the development of visual memory plays a vital role in learning to read. Developing visual memory enables students to recognize and recall letter and word shapes, patterns, and sequences – and in linking words to sounds. Literacy experts have long recognized the significance of visual memory in the reading process and its impact on reading fluency and comprehension. Memory is also essential to comprehension, as a student’s ability to remember relevant background knowledge is key to comprehending text. As Professors Yanling Zhou and Natalie Wong observed, “Children constantly face tasks of differentiating visually similar letters or words. For example, distinguishing “b” from “d,” “a” from “e,” or “book” from “boot” all require visual differentiation. Children’s orthographic knowledge and letter knowledge are causal factors in subsequent reading development.” Professor Linnea C. Ehri developed a theory of “orthographic mapping” – the process of forming letter-sound connections to bond the spellings, pronunciations, and meanings of specific words in memory. She notes: “When we have seen and read a word many times, it is stored in long term memory as a unique letter string and can be read instantly.” And “Various studies indicate that having a visual picture of speech in memory is an important part of a person’s information-processing equipment.” 29 Writing is one excellent way of developing memory. For example, it has been found that writing a word accurately two times is as effective in facilitating word recognition as is reading the same word nine times. This is one of the reasons why Reading Kingdom engages students in a wide variety of writing exercises – from individual words to longer connected texts. The program also makes use a number of very effective techniques for improving student’s memory that include matching exercises, puzzles and visualization tasks. “Modeling can help with decoding, oral reading fluency, reading comprehension strategies, and writing processes.” – Professor Timothy Shanahan Reading is a challenging activity for young students, one that requires the development of many new skills. A tried and proven approach to teaching new skills is modeling. Professor Richard Allington in his book on effective reading instruction observed: “The exemplary teachers in our study routinely gave direct, explicit demonstrations of the cognitive strategies that good readers use when they read. In other words, they modeled the thinking that skilled readers engage in as they attempt to decode a word, self-monitor for understanding, summarize while reading, or edit when composing. [Students] need someone to actually teach it to them — someone who can model and demonstrate.” And Professor Timothy Shanahan wrote: “Modeling can help with decoding, oral reading fluency, reading comprehension strategies, and writing processes, too. Which is why it’s troubling that modeling is used so rarely and so poorly in literacy teaching.” A computer program cannot offer modeling with the detail, finesse, and attention that a human teacher can. However, there are many ways that well-designed software can use the principles of modeling to help teach kids how to read. This is why Reading Kingdom is designed to model best practices for reading in a number of areas. This starts out with introductory reading skills such as left-to-right visual sequencing, sound blending, and spelling and leads up to more complex processes such as comprehension for which Reading Kingdom has developed an innovative comprehension modeling method called “Gleaning Meaning” that guides students in constructing summaries of both narrative and expository stories they have read. Reading Kingdom is an online K-3 reading and writing curriculum that teaches a wide array of literacy skills including phonics, phonemic awareness, spelling, vocabulary, grammar, writing and comprehension. The program makes use of the research from a comprehensive science of reading to implement principles that have been proven to work. You can read the complete article with footnotes, here.

<urn:uuid:22bcea45-f19d-42fd-a1b5-e5f37c25b748>

# college algebra posted by . I need help bad. Don't know which formula to use. Please explain step by step. 2x^2-12x+19. Find the interval where f is increasing and decreasing. • college algebra - If you are confused by formulas, it indicates you don't really understand the concepts. Let's start from scratch. You know the graph is a parabola, which has a vertex. Since the coefficient of x^2 is positive, the vertex is at the minimum value of y. So, left of the vertex y is decreasing, and on the right, it is increasing. So, where is the vertex? y = 2x^2 - 12x + 19 = 2(x^2-6x) + 19 = 2(x^2-6x+9) + 19 - 18 = 2(x-3)^2 + 1 So, the vertex is at (3,1) y decreases for x < 3 y increases for x > 3 • college algebra - Vertex is -b/2a ax^2+by+c=0 or (x2-x1)/(y2-y1)

crawl-data/CC-MAIN-2017-26/segments/1498128320865.14/warc/CC-MAIN-20170626203042-20170626223042-00311.warc.gz

What is glass made of? Glass is a material made primarily from sand. Silica or quartz type of sand is the preferred material in making glass. A small portion of limestone and soda ash is also needed to mix with silica sand to make glass. The purer the silica sand, the clearer will be the glass material output. In case iron is present in the mix, the resulting glass will have a greenish color. In making glass, extremely high temperatures are necessary in order to form the various shapes and sizes of glass. Natural glass is said to be found after a volcanic eruption. The very high temperature of the eruption is basically enough to melt whatever sand is present in a volcano and transform it into a glass-like material. With the origin being a volcanic eruption, natural glass is expected to have a burned or black color. This original glass material was said to be used by ancient people as tips for their spears and other weapons. As time evolved, people have eventually found a way to create glass in different sizes, shapes, and color. Under extremely high temperature, glass components like sand, limestone, and soda ash will basically melt turning the mixture into a type of liquid. This liquid can then be converted back to a solid form as the temperature decreases. At room temperature, the glass material that people are familiar about is in a solid state. The origins of glass though are one in a liquid state because of the melting of sand and other materials. As the sand melts under extreme heat, it can then be manipulated to be formed into different sizes and shapes. This is the part that modern glass-making involves blowing air into the melted liquid to form various shapes and sizes. Once the temperature drops, the glass material will then assume a solid state. Jars, glasses, and figurines for example all started as liquid or melted glass and were later manipulated to become another shape or size as the material reaches room temperature.

<urn:uuid:138354c2-5b0c-41a7-a9eb-dbd5c60bd550>

Geography: Americas, Africa, Middle East, Southeast Asia, Pacific Islands Fatality rate: 1% Cases per year: 100 to 400 million First discovered: 1943 in Japan by Ren Kimura and Susumu Hotta Dengue is spread to people through the bite of an infected mosquito. It is common in more than 100 countries around the world, and about 40% of the world’s population (3 billion people) live in areas with a risk of dengue. There are 4 types of dengue virus, so people could become infected up to 4 times in their lifetime. Dengue epidemics peak during and after rainy seasons due to the increase in mosquito population levels, precipitation and humidity, and air temperatures. 3 out of 4 people infected with dengue do not get sick, but travelers should take precautions. The most common symptom of dengue is fever along with any of the following: nausea and vomiting, rash, aches and pains (bone, joint, muscle, behind the eyes, or headache). Symptoms usually last 2 to 7 days and most people recover after about a week. About 1 in 20 people infected will develop severe dengue, with symptoms such as: stomach pain or tenderness, vomiting, bleeding from the nose or gums, vomiting blood or blood in the stool, and feeling tired or restless. Severe dengue can result in shock, internal bleeding, and even death, and typically appears 24 to 48 hours after the fever has gone away. Protection against dengue includes protecting oneself from mosquito bites by using the correct insect repellent, wearing long pants and long sleeves, and using mosquito netting when sleeping. Once a person has already been infected with dengue virus, they can receive the vaccine. Photo: Dengue virus – Electron micrograph

<urn:uuid:b2669e49-4830-464f-8658-c74c1186b8c4>

Oh, there's a lot of it. This is seaweed. It's pretty humble stuff. But it does have some remarkable qualities. For one, it grows really fast. So the carbon that is part of that seaweed, just a few weeks ago, was floating in the atmosphere as atmospheric CO2, driving all the adverse consequences of climate change. For the moment, it's locked safely away in the seaweed, but when that seaweed rots — and by the smell of it, it's not far away — when it rots, that CO2 will be released back to the atmosphere. Wouldn't it be fantastic if we could find a way of keeping that CO2 locked up long-term, and thereby significantly contributing to solving the climate problem? What I'm talking about here is drawdown. It's now become the other half of the climate challenge. And that's because we have delayed so long, in terms of addressing climate change, that we now have to do two very big and very difficult things at once. We have to cut our emissions and clean our energy supply at the same time that we draw significant volumes of carbon dioxide out of the atmosphere. If we don't do that, about 25 percent of the CO2 we put in the air will remain there, by human standards, forever. So we have to act. This is really a new phase in addressing the climate crisis and it demands new thinking. So, ideas like carbon offsets really don't make sense in the modern era. You know, when you offset something, you say, "I'll permit myself to put some greenhouse gas into the atmosphere, but then I'll offset it by drawing it down." When you've got to both cut your emissions and draw down CO2, that thinking doesn't make sense anymore. And when we're talking about drawdown, we're talking about putting large volumes of greenhouses gases, particularly CO2, out of circulation. And to do that, we need a carbon price. We need a significant price that we'll pay for that service that we'll all benefit from. We've made almost no progress so far with the second half of the climate challenge. It's not on most people's radar. And, you know, I must say, at times, I hear people saying, "I've lost hope that we can do anything about the climate crisis." And look, I've had my sleepless nights too, I can tell you. But I'm here today as an ambassador for this humble weed, seaweed. I think it has the potential to be a big part of addressing the challenge of climate change and a big part of our future. Now, what the scientists are telling us we need to do over the next 80-odd years to the end of this century, is to cut our greenhouse gas emissions by three percent every year, and draw three gigatons of CO2 out of the atmosphere every year. Those numbers are so large that they baffle us. But that's what the scientists tell us we need to do. I really hate showing this graph, but I'm sorry, I have to do it. It is very eloquent in terms of telling the story of my personal failure in terms of all the advocacy I've done in climate change work and in fact, our collective failure to address climate change. You can see our trajectory there in terms of warming and greenhouse gas concentrations. You can see all of the great scientific announcements that we've made, saying how much danger we face with climate change. You can see the political meetings. None of it has changed the trajectory. And this is why we need new thinking, we need a new approach. So how might we go about drawing down greenhouse gases at a large scale? There's really only two ways of doing it, and I've done a very deep dive into drawdown. And I'll preempt my — And I would say this stuff comes up smelling like roses at the end of the day. It does, it's one of the best options, but there are many, many possibilities. There are chemical pathways and biological pathways. So two ways, really, of getting the job done. The biological pathways are fantastic because the energy source that's needed to drive them, the sun, is effectively free. We use the sun to drive photosynthesis in plants, break apart that CO2 and capture the carbon. There are also chemical pathways. They sound ominous, but actually, they're not bad at all. The difficulty they face is that we have to actually pay for the energy that's required to do the job or pay to facilitate that energy. Direct air capture is a great example of a chemical pathway, and people are using that right now to take CO2 out of the atmosphere and manufacture biofuels or manufacture plastics. Great progress is being made, but it will be many decades before those chemical pathways are drawing down a gigaton of CO2 a year. The biological pathways offer us a lot more hope, I think, in the short term. You've probably heard about reforestation, planting trees, as a solution to the climate problem. You know, it's a fair question: Can we plant our way out of this problem by using trees? I'm skeptical about that for a number of reasons. One is just the scale of the problem. All trees start as seeds, little tiny things, and it's many decades before they've reached their full carbon-capture potential. And secondly, if you look at the land surface, you see that it's so heavily utilized. We get our food from it, we get our forestry products from it, biodiversity protection and water and everything else. To expect that we'll find enough space to deal with this problem, I think is going to be quite problematic. But if we look offshore, wee see a solution where there's already an existing industry, and where there's a clearer way forward. The oceans cover about 70 percent of our planet. They play a really big role in regulating our climate, and if we can enhance the growth of seaweed in them, we can use them, I think, to develop a climate-altering crop. There are so many different kinds of seaweed, there's unbelievable genetic diversity in seaweed, and they're very ancient; they were some of the first multicellular organisms ever to evolve. People are using special kinds of seaweed now for particular purposes, like developing very high-quality pharmaceutical products. But you can also use seaweed to take a seaweed bath, it's supposed to be good for your skin; I can't testify to that, but you can do it. The scalability is the big thing about seaweed farming. You know, if we could cover nine percent of the world's ocean in seaweed farms, we could draw down the equivalent of all of the greenhouse gases we put up in any one year, more than 50 gigatons. Now, I thought that was fantastic when I first read it, but I thought I'd better calculate how big nine percent of the world's oceans is. It turns out, it's about four and a half Australias, the place I live in. And how close are we to that at the moment? How many ocean-going seaweed farms do we actually have out there? Zero. But we do have some prototypes, and therein lies some hope. This little drawing here of a seaweed farm that's currently under construction tells you some very interesting things about seaweed. You can see the seaweed growing on that rack, 25 meters down in the ocean there. It's really different from anything you see on land. And the reason being that, you know, seaweed is not like trees, it doesn't have nonproductive parts like roots and trunks and branches and bark. The whole of the plant is pretty much photosynthetic, so it grows fast. Seaweed can grow a meter a day. And how do we sequester the carbon? Again, it's very different from on land. All you need to do is cut that seaweed off — drifts into the ocean abyss, Once it's down a kilometer, the carbon in that seaweed is effectively out of the atmospheric system for centuries or millennia. Whereas if you plant a forest, you've got to worry about forest fires, bugs, etc., releasing that carbon. The key to this farm, though, is that little pipe going down into the depths. You know, the mid-ocean is basically a vast biological desert. There's no nutrients there that were used up long ago. But just 500 meters down, there is cool, very nutrient-rich water. And with just a little bit of clean, renewable energy, you can pump that water up and use the nutrients in it to irrigate your seaweed crop. So I think this really has so many benefits. It's changing a biological desert, the mid-ocean, into a productive, maybe even planet-saving solution. So what could go wrong? Well, anything we're talking about at this scale involves a planetary-scale intervention. And we have to be very careful. I think that piles of stinking seaweed are probably going to be the least of our problems. There's other unforeseen things that will happen. One of the things that really worries me, when I talk about this, is the fate of biodiversity in the deep ocean. If we are putting gigatons of seaweed into the deep ocean, we're affecting life down there. The good news is that we know that a lot of seaweed already reaches the deep ocean, after storms or through submarine canyons. So we're not talking about a novel process here; we are talking about enhancing a natural process. And we'll learn as we go. I mean, it may be that these ocean-going seaweed farms will need to be mobile, to distribute the seaweed across vast areas of the ocean, rather than creating a big stinking pile in one place. It may be that we'll need to char the seaweed — so create a sort of an inert, mineral biochar before we dispatch it into the deep. We won't know until we start the process, and we will learn effectively by doing. I just want to take you to contemporary seaweed farming. It's a big business — it's a six-billion-dollar-a-year business. These seaweed farms off South Korea — you can see them from space, they are huge. And they're increasingly not just seaweed farms. What people are doing in places like this is something called ocean permaculture. And in ocean permaculture, you grow fish, shellfish and seaweed all together. And the reason it works so well is that the seaweed makes the seawater less acid. It provides an ideal environment for growing marine protein. If we covered nine percent of the world's oceans in ocean permaculture, we would be producing enough protein in the form of fish and shellfish to give every person in a population of 10 billion 200 kilograms of high-quality protein per year. So, we've got a multipotent solution here. We can address climate change, we can feed the world, we can deacidify the oceans. The economics of all of this is going to be challenging. We'll be investing many, many billions of dollars into these solutions, and they will take decades to get to the gigaton scale. The reason that I'm convinced that this is going to happen is that unless we get the gas out of the air, it is going to keep driving adverse consequences. It will flood our cities, it will deprive us of food, it will cause all sorts of civil unrest. So anyone who's got a solution to dealing with this problem has a valuable asset. And already, as I've explained, ocean permaculture is well on the road to being economically sustainable. You know, in the next 30 years, we have to go from being a carbon-emitting economy to a carbon-absorbing economy. And that doesn't seem like very long. But half of the greenhouse gases that we've put into the atmosphere, we've put there in the last 30 years. My argument is, if we can put the gas in in 30 years, we can pull it out in 30 years. And if you doubt how much can be done over 30 years, just cast your mind back a century, to 1919, compare it with 1950. Now, in 1919, here in Edinburgh, you might have seen a canvas and wood biplane. Thirty years later, you'd be seeing jet aircraft. Transport in the street were horses in 1919. By 1950, they're motor vehicles. 1919, we had gun powder; 1950, we had nuclear power. We can do a lot in a short period of time. But it all depends upon us believing that we can find a solution. Now what I would love to do is bring together all of the people with knowledge in this space. The engineers who know how to build structures offshore, the seaweed farmers, the financiers, the government regulators, the people who understand how things are done. And chart a way forward, say: How do we go from the existing six-billion-dollar-a-year, inshore seaweed industry, to this new form of industry, which has got so much potential, but will require large amounts of investment? I'm not a betting man, you know. But if I were, I'll tell you, my money would be on that stuff, it would be on seaweed. It's my hero. Thank you. (Applause)

Can seaweed help curb global warming?

# What are 3 characteristics of a proportional relationship? ## What are 3 traits of a proportional relationship? proportional relationship: Two portions are stated to have a proportional relationship in the event that they range in such a manner that one of many portions is a continuing a number of of the opposite, or equivalently if they’ve a continuing ratio. proportion: An equation which states that two ratios are equal. ## What is proportional relationships seventh grade? Proportional relationships are relationships between two variables the place their ratios are equal. Another manner to consider them is that, in a proportional relationship, one variable is all the time a continuing worth occasions the opposite. That fixed is know because the “fixed of proportionality”. ## Does proportional imply equal? A proportional relationship is states that they’re the identical. For instance, 1/2 and 6/12 have a proportional relationship, which suggests they’re the identical. ## Is Y 5x proportional or Nonproportional? Answer. Answer: Yes, any equation of the shape y=mx +c is a proportional relationship. ## What is the unit charge in a proportional relationship known as? Two ratios which are equal are known as a proportion. You can decide if two ratios are proportional by evaluating their unit charges. The unit charge of a proportional relationship is known as the fixed of proportionality. ## Does y 2x 3 symbolize a proportional relationship? Answer: No, it doesn’t represents a proportional relationship. Hence, y = 2x+3 doesn’t symbolize a proportional relationship. ## How are you able to inform if a graph is proportional? The greatest technique to present and clarify direct proportional relationships is by graphing two units of associated portions. If the relation is proportional, the graph will type a straight line that passes by way of the origin. ## Is Y 19x 5 proportional? But the equation y = 19x – 5 given within the query reveals the y intercept of (-5). Therefore, this equation can’t present the proportionality relation between x and y. It’s non proportional. ## Is Y =- 9x proportional or Nonproportional? Is y=-9x proportional oor nonproportional. y= -9x is proportional. This reply has been confirmed as right and useful. ## What is proportional time? By no matter ratio one amount adjustments, the opposite adjustments in the identical ratio. For instance, allow us to say that the gap you journey is proportional to the time. This implies that if you happen to journey twice as lengthy, you’ll go twice as far. If you journey 3 times as lengthy, you’ll go 3 times as far. ## Is time immediately proportional to distance? – When the velocity is fixed, time is immediately proportional to distance. – When time is fixed, then velocity is immediately proportional to distance. – When the gap is fixed, then velocity is inversely proportional to time. ## Is proportional to image? The image used to indicate the proportionality is’∝’. For instance, if we are saying, a is proportional to b, then it’s represented as ‘a∝b’ and if we are saying, a is inversely proportional to b, then it’s denoted as ‘a∝1/b’. ## What is an instance of immediately proportional? The time period direct proportion implies that two (or extra) portions improve or lower in the identical ratio. For instance, within the purple paint combination, the ratio of blue to pink of 4:3. This doesn’t essentially imply we each combination would have 4 pink and three blue cans. It does imply we’d have 4 pink for each 3 blue cans. ## What is a direct proportion in math? Direct proportion is the connection between two variables whose ratio is the same as a continuing worth. In different phrases, direct proportion is a state of affairs the place a rise in a single amount causes a corresponding improve within the different amount, or a lower in a single amount ends in a lower within the different amount. ## What is the method for proportion? Ratios and Proportions – Proportions – In Depth. A proportion is solely a press release that two ratios are equal. It might be written in two methods: as two equal fractions a/b = c/d; or utilizing a colon, a:b = c:d. The following proportion is learn as “twenty is to 25 as 4 is to 5.” ## What are the three sorts of proportion? Types of Proportions • Direct Proportion. • Inverse Proportion. ## What are the primary and final numbers in a proportion? In a proportion corresponding to 3:8 = 9:24, the primary and the final phrases (the surface phrases) are known as the EXTREMES. In different phrases, the numerator of the primary ratio and the denominator of the second are known as the extremes. The second and third phrases (the within phrases) are known as the MEANS. ## How do you name the 4 numbers in a proportion? The numbers in a proportion are known as the phrases: the first, the 2nd, the third, and the 4th. We say that the first and the third are corresponding phrases, as are the 2nd and the 4th. corresponding phrases are additionally proportional. ## Are 30 40 45 and 60 are in proportion? Ratio of 30 to 40 = = 3: 4. Ratio of 45 to 60 = = 3: 4. Therefore, 30, 40, 45, and 60 are in proportion. ## How have you learnt if a quantity is in proportion? If a : b :: b : c, then we are saying {that a}, b, c are in continued proportion, then c is the third proportional of a and b. Also, b is known as the imply proportional between a and C…. 1. Determine if 8, 10, 12, 15 are in proportion. 2. Check if 6, 12, 24 are in proportion. 3. Find the fourth Proportional to 12, 18, 20. 20 : 28 25% ## What are 3 ratios which are equal to 7 6? Answer: 14 : 12 , 21 : 18 and 28 : 24 . ## What is 5/8 as a ratio? Actually 58 is a ratio. All fractions are expressions of ratios. The second a part of 5:8 is the denominator. This implies that the primary merchandise has 5 elements for each 8 elements of the second merchandise. ## What is the ratio of three to five? 3 : 5 = ? : 40. (3 out of 5 is what number of out of 40?) “5 goes into 40 eight occasions. Eight occasions 3 is 24.”

crawl-data/CC-MAIN-2022-33/segments/1659882572192.79/warc/CC-MAIN-20220815145459-20220815175459-00631.warc.gz

## Enter the value for x that makes the equation 2(x – 3) + 21 = –3 true. PLEASE I NEED THE ANSWER Question Enter the value for x that makes the equation 2(x – 3) + 21 = –3 true. in progress 0 2 weeks 2022-01-08T10:07:39+00:00 2 Answers 0 views 0 1. 2(x-3)+21=-3 Distribute the 2 2x-6+21=-3 Combine like terms 2x+15=-3 Subtract 15 on both sides 2x=-18 Isolate x, so divide both sides by 2 X=-9

crawl-data/CC-MAIN-2022-05/segments/1642320304749.63/warc/CC-MAIN-20220125005757-20220125035757-00692.warc.gz

Gold rush worksheets can be a fun way to approach history. This board game about the California Gold Rush is a thrilling race across the Wild West! Sutter's Mill is where the California Gold Rush first began! Your student will read about the first discovery of gold with this history worksheet. Dive into Native American history with your third grader! He'll get an introduction to Native American symbols and tradition as he colors his own shield. Teach your student a bit about how our country was formed with this worksheet about the Gadsden Flag. Help your fourth grader learn about the legislative process with this worksheet, which challenges him to follow the path of a law in the making. We have four different time zones here in America, but do you know how to use them? This worksheet will help your student learn about U.S. time zones. Did you know there are 24 different time zones across the globe? Have your 4th grade student practice his time zone calculations with these word problems. There's tons of trivia in this social studies worksheet about America's most famous home. Welcome to the Wild West! Learn about the Gold Rush and the founding of San Francisco, with this worksheet about the talented and versatile Sam Brannan. The history of the American West meets the history of fashion! Your child will learn both with this worksheet about Levi Strauss, creator of Levi's Jeans.

<urn:uuid:af3dd7b7-1d4c-41c6-bd54-ee0c8f65db8d>

Optical lens design Optical lens design is the process of designing a lens to meet a set of performance requirements and constraints, including cost and manufacturing limitations. Parameters include surface profile types (spherical, aspheric, holographic, diffractive, etc.), as well as radius of curvature, distance to the next surface, material type and optionally tilt and decenter. The process is computationally intensive, using ray tracing or other techniques to model how the lens affects light that passes through it. Performance requirements can include: - Optical performance (image quality): This is quantified by various metrics, including encircled energy, modulation transfer function, Strehl ratio, ghost reflection control, and pupil performance (size, location and aberration control); the choice of the image quality metric is application specific. - Physical requirements such as weight, static volume, dynamic volume, center of gravity and overall configuration requirements. - Environmental requirements: ranges for temperature, pressure, vibration and electromagnetic shielding. Design constraints can include realistic lens element center and edge thicknesses, minimum and maximum air-spaces between lenses, maximum constraints on entrance and exit angles, physically realizable glass index of refraction and dispersion properties. Manufacturing costs and delivery schedules are also a major part of optical design. The price of an optical glass blank of given dimensions can vary by a factor of fifty or more, depending on the size, glass type, index homogeneity quality, and availability, with BK7 usually being the cheapest. Costs for larger and/or thicker optical blanks of a given material, above 100–150 mm, usually increase faster than the physical volume due to increased blank annealing time required to achieve acceptable index homogeneity and internal stress birefringence levels throughout the blank volume. Availability of glass blanks is driven by how frequently a particular glass type is made by a given manufacturer, and can seriously affect manufacturing cost and schedule. Lenses can first be designed using paraxial theory to position images and pupils, then real surfaces inserted and optimized. Paraxial theory can be skipped in simpler cases and the lens directly optimized using real surfaces. Lenses are first designed using average index of refraction and dispersion (see Abbe number) properties published in the glass manufacturer's catalog and though glass model calculations. However, the properties of the real glass blanks will vary from this ideal; index of refraction values can vary by as much as 0.0003 or more from catalog values, and dispersion can vary slightly. These changes in index and dispersion can sometimes be enough to affect the lens focus location and imaging performance in highly corrected systems. The lens blank manufacturing process is as follows: - The glass batch ingredients for a desired glass type are mixed in a powder state, - the powder mixture is melted together in a furnace, - the fluid is further mixed while molten to maximize batch homogeneity, - poured into lens blanks and - annealed according to empirically determined time-temperature schedules. The glass blank pedigree, or "melt data", can be determined for a given glass batch by making small precision prisms from various locations in the batch and measuring their index of refraction on a spectrometer, typically at five or more wavelengths. Lens design programs have curve fitting routines that can fit the melt data to a selected dispersion curve, from which the index of refraction at any wavelength within the fitted wavelength range can be calculated. A re-optimization, or "melt re-comp", can then be performed on the lens design using measured index of refraction data where available. When manufactured, the resulting lens performance will more closely match the desired requirements than if average glass catalog values for index of refraction were assumed. Delivery schedules are impacted by glass and mirror blank availability and lead times to acquire, the amount of tooling a shop must fabricate prior to starting on a project, the manufacturing tolerances on the parts (tighter tolerances mean longer fab times), the complexity of any optical coatings that must be applied to the finished parts, further complexities in mounting or bonding lens elements into cells and in the overall lens system assembly, and any post-assembly alignment and quality control testing and tooling required. Tooling costs and delivery schedules can be reduced by using existing tooling at any given shop wherever possible, and by maximizing manufacturing tolerances to the extent possible. A simple two-element air-spaced lens has nine variables (four radii of curvature, two thicknesses, one airspace thickness, and two glass types). A multi-configuration lens corrected over a wide spectral band and field of view over a range of focal lengths and over a realistic temperature range can have a complex design volume having over one hundred dimensions. Lens optimization techniques that can navigate this multi-dimensional space and proceed to local minima have been studied since the 1940s, beginning with early work by James G. Baker, and later by Feder, Wynne, Glatzel, Grey and others. Prior to the development of digital computers, lens optimization was a hand-calculation task using trigonometric and logarithmic tables to plot 2-D cuts through the multi-dimensional space. Computerized ray tracing allows the performance of a lens to be modelled quickly, so that the design space can be searched rapidly. This allows design concepts to be rapidly refined. In most cases the designer must first choose a viable design for the optical system, and then numerical modelling is used to refine it. The designer ensures that designs optimized by the computer meet all requirements, and makes adjustments or restarts the process when they do not. - Fischer, Robert E.; Tadic-Galeb, Biljana; Yoder, Paul R. (2008). Optical System Design (2nd ed.). New York: McGraw-Hill. pp. 8, 179–198. ISBN 0-07-147248-7. - "Modulation Transfer Function". - D.P. Feder, "Automatic Optical Design," Appl. Opt. 2, 1209–1226 (1963). - C. G. Wynne and P. Wormell, "Lens Design by Computer," Appl. Opt. 2:1223–1238 (1963). - "Dr. Erhardt Glatzel (Biography)". The Zeiss Historica Society. Archived from the original on January 27, 2013. Retrieved July 21, 2013. - Grey, D.S., "The Inclusion of Tolerance Sensitivities in the Merit Function for Lens Optimization", SPIE Vol. 147, pp. 63–65, 1978. - Fischer (2008), pp. 171–5. - Smith, Warren J., Modern Lens Design, McGraw-Hill, Inc., 1992, ISBN 0-07-059178-4 - Kingslake, Rudolph, Lens Design Fundamentals, Academic Press, 1978 - Shannon, Robert R., The Art and Science of Optical Design, Cambridge University Press, 1997.

<urn:uuid:6709612d-b96d-490f-84cc-d8156c0ae55e>

# Picture Subtraction Worksheets How Using Pictures Can Help Learn to Subtract? There is a saying that a picture is worth a thousand words, and that does not only mean in literature. Pictures are a great learning method to make the students understand the concept of basic mathematics. Especially if you consider the arithmetic operations where everything is related to our daily lives. As a teacher, you want your students to understand the concept you’re teaching them perfectly. However, to do so, you need to devise strategies that could enable them to understand the concepts perfectly. In hindsight, with pictures, you will be able to make the students visualize actual scenarios. Furthermore, the concept of mental math comes into the picture, which will not only improve the minds of the students but make them better in the concept. • ### Basic Lesson Demonstrates subtraction by crossing out items. Start with 7 strawberries. Cross out 3 strawberries. How many strawberries do you have left? • ### Intermediate Lesson This lesson demonstrates subtraction through the use of a number line. Use the number line to find the answer. Start at the number 5, then count three numbers to the left. • ### Independent Practice 1 Students cross out pictures to complete subtraction problems. The answers can be found below. There are 4 dinosaurs. Cross out 3 dinosaurs. How many dinosaurs are left? • ### Independent Practice 2 Students refer to a number line to complete subtraction problems. • ### Homework Worksheet Reviews the skills within the unit and provides 8 practice problems. • ### Skill Quiz Incorporates picture subtraction and number line subtraction math problems to help assess the unit. Start with 9 pineapples. Cross out 2 pineapples. How many pineapples do you have left? • ### Homework and Quiz Answer Key Answers for the homework and quiz. Answers for the lesson and practice sheets. • ### Basic Lesson Demonstrates a word problem mixed with a few visual cues. The shop has 80 printers. People buy 30. How many are left? • ### Intermediate Lesson This lesson demonstrates subtraction using larger numbers than the basic lesson. The shop has 75 paintings. People buy 30. How many are left? • ### Independent Practice 1 Takes 4 scenarios and makes 5 practice out of each scenario. The answers can be found below. • ### Independent Practice 2 Another 20 practice problems. The answer key is below. • ### Homework Worksheet Reviews the skills within the unit and provides 12 practice problems. • ### Skill Quiz 10 visual subtraction math problems to help assess the unit. A math scoring matrix is included. • ### Homework and Quiz Answer Key Answers for the homework and quiz.

crawl-data/CC-MAIN-2024-22/segments/1715971058147.77/warc/CC-MAIN-20240520015105-20240520045105-00349.warc.gz

Ok so today we are going to talk about labels First off, what are labels? - a classifying phrase or name applied to a person or thing, especially one that is inaccurate or restrictive “inaccurate or restrictive”?? That’s from an online dictionary Basically when you Google “label meaning” Cambridge dictionary describes a label as - a word or a phrase that is used to describe the characteristics or qualities of people, activities, or things, often in a way that is unfair Now a label is also “unfair”??? So guess what? A label is inaccurate, restrictive and unfair So that’s it right? Well that’s the problem You see, you’ve probably gave a label to someone before Which means it could have been inaccurate, restrictive and unfair A parent might label their child as “naughty” A teacher might label a student as “stupid” A girl might think a stranger looks “like a criminal” A guy might think the neighbour is “noisy” The list goes on And you know what’s worse than labelling your friend, family, child, colleague, staff etc something that is inaccurate, restrictive and unfair? You might label yourself in an inaccurate, restrictive and unfair manner I’m not joking Isn’t putting yourself down much worse than labelling others? And it gets worse! Let me share an experiment So I don’t know when but some psychologists gave an assessment to a class of say 40 students And they got the top 5 and bottom 5 students What happened is that they told the teacher the top 5 are the bottom 5 and the bottom 5 are the top 5 Essentially, they swap the results So a year later, they did an assessment for the students and found that the original top 5, told to the teacher as the bottom 5, are now in the bottom 5 and vice versa So what happened? The teachers might have have had a label of the top 5 and bottom 5 students When the top 5 (told to her as bottom 5) want to ask a question, the teacher might have felt irritated cause she might have labelled the bottom 5 as stupid, naughty or even troublemakers And when the bottom 5 (told to her as top 5) want to ask a question, the teacher might have shown more attention or give praise cause she might have labelled the top 5 as smart, curious or even proactive Of course, I’m adding my possible opinions for the study and what might have happened And it’s true isn’t it? When you think a particular friend is being all whiny, you start to treat him or her differently You rather get away from this person, or even throw a temper at him or her Similarly, if you label someone as good and kind, she or he will get your best treatment and you might even grow from affectionate So how now brown cow? What can we do as individuals? Can we completely stop negative labelling? Or remove labelling totally?? Unfortunately, probably not But we can make a step towards positive change And it starts with you By having a more understanding way of interacting with people And accepting yourself as who you are and making tiny changes For example, yea you might get first impressions from others but remember that you cannot judge a book by its cover, and then have a chat with an open mind When you see a child screaming and yelling in a public place, don’t label the child as naughty (truthfully, the parents might not have been giving the right attention of “praise” and thus the child seeks the wrong attention of “scold”) Especially your own kids!! Don’t think they have your genes and therefore must be good in math! With this, I hope you gain more awareness of yourself too How you label yourself, you portray that to others If you think you are worthless, you will show yourself to be such, and people will see that Kinda like a self fulfilling prophesy but that’s a topic for another day I’m not telling you to lie to yourself! If you are really committing a mistake, take steps to change it And become a better person one step at a time 🙂

<urn:uuid:807e4d61-bc75-4641-a5cb-da9460285b13>

# Matrix Norm ## 1 Definition # Consider a number field $$K$$ which is either real or complex. The matrix norm is a function $$\| \cdot \| : K^{m \times n} \to \RR$$ that satisfies the following properties: For all scalars $$\alpha \in K$$ and for all matrices $$A,b \in K^{m \times n}$$, • $$\|A\|\geq 0$$ • $$\|A\| = 0 \Longleftrightarrow A = 0_{m,n}$$ • $$\|\alpha A\|= |\alpha| \|A\|$$ • $$\|A+B\|\leq \|A\|+\|B\|$$ Additionally, in the case of square matrices, some (but not all) matrix norms satisfy the following sub-multiplicative condition. • $$\|AB\|\leq \|A\|\|B\|$$ A matrix norm that satisfies this additional property is called a sub-multiplicative norm ## 2 Operator Norm # Suppose a vector norm $$\| \cdot \|$$ on $$K^m$$ and $$K^n$$ is given, then we define the corresponding induced norm or operator norm on the space $$K^{m\times n}$$ as follows: \begin{align} \|A\| &=\sup \left\{ \|Ax\|: x\in K^n, \|x\|=1 \right\}\\ &=\sup \left\{ \|Ax\|: x\in K^n, \|x\|\leq 1 \right\}\\ &=\sup \left\{ \frac{\|Ax\|}{\|x\|}: x\in K^n, x\neq 0 \right\} \end{align} The last equality is usually reformed and used as an inequality: $\|Ax\| \leq \|A\|\|x\|$ Any induced operator norm is a sub-multiplicative matrix norm. This follows from: $\|ABx\|\leq \|A\|\|Bx\|\leq \|A\|\|B\|\|x\|$ and $\max_{\|x\|=1} \|ABx\| = \|AB\|$ ## 3 Frobenius Norm # Frobenius norm treats an $$m \times n$$ matrix as a vector of size $$m \cdot n$$: $\|A\|_F = \sqrt{\langle A,A\rangle _{F}}$ where $$\langle A,A\rangle_{F}$$ is the Frobenius inner product, defined as $\langle A,A\rangle_{F} = \sum_{i,j} \overline{A_{ij}}B_{ij} = \tr \left( \overline{A^T}B \right) = \tr \left( A^{\dagger}B \right)$

crawl-data/CC-MAIN-2023-50/segments/1700679100308.37/warc/CC-MAIN-20231201215122-20231202005122-00416.warc.gz

How did tiny Mercury get its name? Mercury was named for the Roman messenger god who traveled rapidly on his winged sandals. From the vantage point of Earth, the planet Mercury travels swiftly across the face of the Sun. The smallest planet, Mercury, is the planet closest to the Sun. Because Mercury is so close to the Sun, it is difficult to observe from Earth, even with a telescope. However, the Mariner 10 spacecraft, shown in , visited Mercury from 1974 to 1975. The MESSENGER spacecraft has been studying Mercury in detail since 2005. The craft is currently in orbit around the planet, where it is creating detailed maps. MESSENGER stands for Mercury Surface, Space Environment, Geochemistry and Ranging. (a) Mariner 10 made three flybys of Mercury in 1974 and 1975. (b) A 2008 image of compiled from a flyby by MESSENGER. shows, the surface of Mercury is covered with craters, like Earth’s Moon. Ancient impact craters means that for billions of years Mercury hasn’t changed much geologically. Also, with very little atmosphere, the processes of weathering and erosion do not wear down structures on the planet. Mercury is covered with craters, like Earth’s Moon. MESSENGER has taken extremely detailed pictures of the planet’s surface. There are many images, movies, and activities on the MESSENGER site: Short Year, Long Days Mercury is named for the Roman messenger god, who could run extremely quickly, just as the planet moves very quickly in its orbit around the Sun. A year on Mercury — the length of time it takes to orbit the Sun — is just 88 Earth days. Despite its very short years, Mercury has very long days. A day is defined as the time it takes a planet to turn on its axis. Mercury rotates slowly on its axis, turning exactly three times for every two times it orbits the Sun. Therefore, each day on Mercury is 57 Earth days long. In other words, on Mercury, a year is only a Mercury day and a half long! Mercury is close to the Sun, so it can get very hot. However, Mercury has virtually no atmosphere, no water to insulate the surface, and it rotates very slowly. For these reasons, temperatures on the surface of Mercury vary widely. In direct sunlight, the surface can be as hot as 427°C (801°F). On the dark side, or in the shadows inside craters, the surface can be as cold as -183°C (-297°F)! Although most of Mercury is extremely dry, scientists think there may be a small amount of water in the form of ice at the poles of Mercury, in areas that never receive direct sunlight. A Liquid Metal Core shows a diagram of Mercury’s interior. Mercury is one of the densest planets. It’s relatively large, liquid core, made mostly of melted iron, takes up about 42% of the planet's volume. Mercury contains a thin crust, a mantle, and a large, liquid core that is rich in iron. Mercury appears to be moving rapidly because it's so close to the Sun. Mercury has short years, just 88 Earth days, but long days, about 57 Earth days. Mercury is extremely hot and has a liquid metal core. Use this resource to answer the questions that follow. What is the location and size of Mercury? Why do the temperatures on Mercury vary so much? Why was Mercury named for the messenger god Mercury? Why is Mercury so dense? What are the features of its core? Why is the landscape similar to our Moon's? Why hasn't the surface of Mercury changed over it's history, except for the addition of more impact craters? How can ice be found on such a hot planet? Why does it take so much rocket fuel to send a spacecraft to Mercury? What do probes to Mercury reveal? Want to know more about Mercury? See Why is a year on Mercury only 88 days long? Why is Mercury mostly really hot, but very cold in spots? Think about the formation of the solar system. Why is Mercury the densest planet?

<urn:uuid:dd23bf56-a332-4df2-915a-20c0ab5ffa7a>

Today, many scientists believe we are on the cusp of a sixth mass extinction which could wipe out most life on Earth as we know it. Here are seven signs that they could be right. Image of Australian wildfires from space, via NASA A mass extinction happens when over 75 percent of all species on the planet die in a period of less than two million years. That may sound long to you, but it's the blink of an eye in geologic time. There have been five mass extinctions on Earth over the past 540 million years, sometimes caused by catastrophic disasters, and sometimes by quiet, insidious events like invasive species taking over the planet. 7. Earth Is Bubbling with Super Volcanoes Yellowstone Park in the United States is actually a volcano caldera, a thin cork of earth that sits on top of a massive cache of broiling magma. And this super-volcano could blow any time. The last time Earth witnessed an explosion of this size was in 1812, when Mount Tambora in Indonesia erupted so profusely that the Earth's climate cooled for several years afterwards. Even more frightening is the prospect that another kind of super volcano, called a large igneous province (LIP), could become active sometime in the future. A now-inactive LIP, called the Siberian Traps, erupted 250 million years ago. It spewed so much sulfur, carbon other greenhouse gases into the air that the Earth experienced a climate change catastrophe, vacillating wildly between extreme heat and cold until 95 percent of all life had died. This mass extinction was so bad it's been nicknamed "the Great Dying" by geologists. Yellowstone is not a LIP, but if it explodes into a super eruption, the damage will be incredible. Super volcanoes are an ever-present threat, that have haunted the Earth for millions of years. 6. Invasive Species Are Everywhere On Earth, humans have aggressively invaded every continent except Antarctica, swelling our population to over 7 billion individuals and eating everything in sight. Like rats and cockroaches, we are the ultimate invasive species, pushing many creatures out of their native habitats — which could, ultimately, kill those creatures on a huge scale. Our population could grow a lot bigger before humans are endangered, but that doesn't mean we wouldn't harm other species. About 359 million years ago, 75 percent of all species on Earth died during the Devonian mass extinction. Geologist Alicia Stigall has evidence that this horrific slaughter was the result of invasive species like sharks (yes, there were sharks hundreds of millions of years ago) aggressively eating all the food in every environment — slowly starving all the creatures who depended on local food sources and couldn't move to new feeding regions. 5. Climate Change The Arctic ice cap is shrinking. Temperatures are rising. Scientists across many disciplines and countries are united in their belief that the climate on Earth is getting hotter. The good news is that humans might not be the only cause of this climate change — the planet has suffered through dramatic shifts in temperature many times over its history. The bad news is that pretty much every time that happens there's a mass extinction. The Great Dying was caused by climate change. The very first mass extinction 540 million years ago, called the Ordovician Extinction, was set off by a rapid ice age followed by a rapid greenhouse period. Another mass extinction, at the end of the Triassic, was set in motion by an undersea super volcano and massive wildfires (like the ones in Australia, pictured from space in the image at the top of this post) that smothered the planet in smoke and ash. The meteorite that smashed into the planet before the dinosaurs were wiped out in a mass extinction? Nope, didn't kill those big guys with fire. It killed them by throwing so much debris into the atmosphere that the climate changed. Most geologists agree that when the climate changes, mass extinctions follow. 4. Ocean Acidification Acid levels in the Earth's oceans are going up, which is what's killing all those reefs and making life hard for shellfish. Ocean acidification is also one major reason that the Great Dying was so, well, great. It was also a major part of the Triassic mass extinction 200 million years ago, which wiped out 80 percent of the planet's species — especially in the oceans. When the waters are too acidic, calcium levels go down. That means shelled creatures simply can't build their shells, and they die even before they have a fighting chance. When shelled creatures die, the predators who feed on them also die. And the more dead bodies you've got in the ocean, the more acidic everything gets. If Earth's oceans continue to become more acidic, mass extinction could be next. 3. Extinctions Are Happening At A Higher Than Average Rate Extinctions are normal. In fact, statisticians who study extinction have figured out a typical "background extinction rate," which is the normal number of creatures who are going extinct at any given time. So a mass extinction is like a big statistical spike of death sticking up far over that background rate. And, unfortunately, there is a lot of evidence that the extinction rate we've experienced over the past 500 years is above the typical rate. No, this rate is nowhere near mass extinction levels. But it is going up. Which is exactly what you'd expect to see at the beginning of a mass extinction. 2. All the Megafauna Are Dead One way scientists figure out rates of extinction is by looking at the diversity of fossils. Based on this evidence, they can figure out how many creatures and plants were alive at a given time, plus how quickly (or slowly) they disappeared from the fossil record. In recent fossil records, from the past 50,000 years, we can easily see a steep decline in species diversity. The Earth was recently home to many species of so-called megafauna, from mastodons and giant wallabies, to giant sloths, and today they are entirely gone. When you see an entire category of creatures wink out that quickly (in geological time), it suggests more than just typical extinction patterns. 1. Amphibians Are Dying Out Today, we are witnessing another giant group of species going extinct so rapidly we can actually measure it in human time, rather than geologic time. Amphibians, especially frogs, are dying out at such a fast pace that some have called the twenty-first century a time of "biodiversity crisis." Most have been felled by a fast-spreading, deadly fungus that kills whole communities of frogs in weeks. It's likely the fungus has reached pandemic proportions because frogs are being forced out of their habitats, and coming into contact with new species they might never have seen otherwise. Just as human pandemics spread more quickly due to travel, amphibian pandemics are spread when frogs move into a new area and infect previously unexposed communities. The more we lose our animal diversity, the closer we get to a world dominated by invasive species. And that scenario really didn't end well in the Devonian mass extinction. It probably won't end well for us, either. Still, as I explain in my book, Scatter, Adapt, and Remember: How Humans Will Survive a Mass Extinction, there is hope. These are early signs of a possible mass extinction, and we still have plenty of time to do something about it. We can curb fossil fuel emissions to prevent climate change from getting worse, and we can preserve biodiversity by maintaining natural areas where animals won't be edged out by human settlements. As for megavolcanos and meteorite impacts? Well, that's going to be a little harder. But it's not impossible. We can't bring the mastodons back, but we can still prevent most of the species around us (including humans) from dying out. If you want to know more about mass extinctions, you can learn about it in my new book, Scatter, Adapt and Remember: How Humans Will Survive a Mass Extinction. Also, I'll be on book tour this month! You can see me in tomorrow night in Phoenix, at Changing Hands Bookstore. That's followed by appearances in Seattle (this Wednesday night!), Chicago, Atlanta, San Francisco, and Berkeley. Click here for dates and places! Primary sources linked within the post. A previous version of this io9 flashback was published in October 2012.

<urn:uuid:e18ffcbe-93fa-4df1-bb2f-69ef4db1a042>

# 4.2: Marginal Analysis $$\newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$$ $$\newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$$$$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}[1]{\| #1 \|}$$ $$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}[1]{\| #1 \|}$$ $$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$ ### Learning Objectives • Explain the importance of marginal analysis in economics • Give examples of marginal cost and marginal benefit The budget constraint framework helps to illustrate that most choices in the real world are not about getting all of one thing or all of another—we rarely decide “all burgers” or “all bus tickets.” Options usually fall somewhere on a continuum, and the choice usually involves marginal decision-making and marginal analysis. Marginal decision-making means considering a little more or a little less than what we already have. We decide by using marginal analysis, which means comparing the costs and benefits of a little more or a little less. It’s natural for people to compare costs and benefits, but often we look at total costs and total benefits, when the best choice requires comparing how costs and benefits change from one option to another. In short, you might think of marginal analysis as “change analysis.” Marginal analysis is used throughout economics. This subtle concept is easier to grasp with examples. ## Marginal Cost Figure 1. Charm Bracelet. What is the marginal cost of getting more silver heart charms? Should you buy just one charm for $4, or all of them for$12? Generally speaking, marginal cost is the difference (or change) in cost of a different choice. From a consumer’s point of view, marginal cost is the additional cost of one more item purchased. From a business’s point of view, marginal cost is the additional cost of one more item produced. Suppose you typically spend a week at the beach for vacation, but this year you earned an annual bonus from your job. Should you rent a beach house for one week or two? A one-week rental costs $2,000. A two-week rental costs$3,600. Holding everything else constant, which option is better? If you stay for two weeks, the cost is significantly higher: $3,600 versus$2,000. But consider the cost by week. The first week costs $2,000. The difference in cost between one week and two is$3,600 – $2,000, or$1,600. Thus, while the marginal cost of the first week’s rental is $2,000, the marginal cost of the second week’s rental is$1,600. This illustrates the key rule of marginal analysis: Marginal cost = the change in total cost from one option to another. Consider another example. Imagine that you’re out getting ice cream with your friends or family. You can choose whether to buy one, two, or three scoops of ice cream. One scoop costs $3.00, two scoops cost$5.00, and three scoops cost $7.00. This information is shown in the following table. Scoops of Ice Cream 1 2 3 Total Cost$3 $5$7 What is the marginal cost of each scoop of ice cream? The marginal cost of the first scoop of ice cream is $3.00 because you have to pay$3.00 more to get one scoop of ice cream than you do to get zero scoops of ice cream. The marginal cost of the second scoop of ice cream is $2.00 because you only need to pay two more dollars to get two scoops than you need to pay to get one scoop. The marginal cost of the third scoop is also$2.00 because you would need to pay an additional two dollars to get that third scoop. ### Try It These next questions allow you to get as much practice as you need, as you can click the link at the top of the first question (“Try another version of this question”) to get a new set of questions. Practice until you feel comfortable doing the questions and then move on. [ohm_question sameseed=1]92669-92670-92671[/ohm_question] ### Glossary [glossary-page][glossary-term]marginal analysis: [/glossary-term] [glossary-definition]examination of decisions on the margin, meaning comparing costs of a little more or a little less[/glossary-definition][glossary-term]marginal benefit: [/glossary-term][glossary-definition]the difference (or change) in what you receive from a different choice[/glossary-definition][glossary-term]marginal cost: [/glossary-term] [glossary-definition]the difference (or change) in cost of a different choice[/glossary-definition][/glossary-page]

crawl-data/CC-MAIN-2022-27/segments/1656103334753.21/warc/CC-MAIN-20220627134424-20220627164424-00452.warc.gz

# 102 in Words: How to Spell One Hundred Two and Solved Examples The number 102 is written as One Hundered Two in English words. It is a natural number and is also a odd number. You can use One Hundered Two in various ways. For instance, if you completed 102 levels in a video game, you could say, “I’ve finished One Hundered Two levels!” Additionally, 102 possesses interesting mathematical properties, such as not being perfectly divisible by 2. To know more about 102 in words, you can read the given article. ## 102 in Words in English One Hundered Two is the accepted way to express 102 in written English. We employ this form in both everyday communication and mathematical expressions. Here’s an example: “The test score required for admission is One Hundered Two.” ## How to Convert 102 in Words? Similar to other numbers, we use the alphabet to spell out 102. Here’s a how you can spell 102 in words: • Hundreds Place: Nine • Tens Place: One • Ones Place: (Empty) (Since 102 falls between 90 and 100, the ones place is typically not used) ## Facts About the Number 102 • Odd Number: As mentioned before, 102 is not divisible by 2, making it an odd number. • Composite Number: Unlike 83 (a prime number), 102 is a composite number. This means it has more than two factors (1, 7, 13, and 102 itself). ## Solved Examples on 102 in Words Let’s practice using One Hundered Two word problems: Question 1: There are 102 marbles in a bag. If you distribute them equally among 3 friends, how many marbles will each friend receive? Solution: • Divide the total by the number of people: 102 marbles / 3 friends = 30 marbles/person • Each friend will get 30 marbles. Question 2: A rectangular bookshelf is 102 centimeters tall and 25 centimeters wide. What is the area of the bookshelf? Solution: • Length of the bookshelf: One Hundered Two centimeters • Width of the bookshelf: 25 centimeters • Area = Length x Width = 102 centimeters x 25 centimeters = 2275 square centimeters • The area of the bookshelf is 2275 square centimeters. ## 35 to 102 Numbers in Words You can find resources online that list all numbers 35 to 102 in words. For now, you can focus on practicing writing 83! ## FAQs How do you spell 102 in English words? 102 is spelled as “One Hundered Two”. What is the meaning of 102 in English? 102 means One Hundered Two, which is one more than ninety. What is 102st in words? 102st is written as “Ninety-First”. This was all about the “102 in Words”.  For more such informative blogs, check out our Study Material Section, or you can learn more about us by visiting our Indian exams page

crawl-data/CC-MAIN-2024-33/segments/1722640365107.3/warc/CC-MAIN-20240803091113-20240803121113-00364.warc.gz

- deg C - degree C - The Celsius scale, already widely used in Europe, replaced the Fahrenheit scale in most countries during the mid-to-late 20th century, although Fahrenheit remains the official scale of the United States, Cayman Islands and Belize. Although initially defined by the freezing point of water (and later the melting point of ice), the Celsius scale is now officially a derived scale, defined in relation to the Kelvin temperature scale. Zero on the Celsius scale (0 °C) is now defined as the equivalent to 273.15 K, with a temperature difference of 1 deg C equivalent to a difference of 1 K, meaning the unit size in each scale is the same. This means that 100 °C, previously defined as the boiling point of water, is now defined as the equivalent to 373.15 K. The Celsius scale is an interval system but not a ratio system, meaning it follows a relative scale but not an absolute scale. This can be seen because the temperature interval between 20 °C and 30 °C is the same as between 30 °C and 40 °C, but 40 °C does not have twice the air heat energy of 20 °C. A temperature difference of 1 deg C is the equivalent of a temperature difference 1.8°F. The Celsius scale is named after the Swedish astronomer Anders Celsius (1701-1744). In 1742, Celsius created a temperature scale wherein 0 degrees was the boiling point of water and 100 degrees the freezing point. Around this time other physicists independently developed a similar scale but reversed, such that 0 degrees was the melting point of ice and 100 degrees the boiling point of water. This new ‘forward’ scale was widely adopted across continental Europe, generally being referred to as the centigrade scale. The scale was officially named as ‘The Celsius scale’ in 1948 to prevent confusion with the use of centigrade as an angular measurement. - Absolute Zero, -273.15 °C - Melting point of ice, 0 °C (actually -0.0001 °C) - Warm summer's day in a temperate climate, 22 °C - Normal human body temperature, 37 °C - Boiling point of water at 1 atmosphere, 99.9839 °C

<urn:uuid:5fa1758f-e344-4689-ae3d-4e8f1d28117b>

Gender stereotypes are not as prevalent in television with adults; yet, children are still confined to the roles of their specific gender. Commercials, and other forms of media continue to show men, women, boys and girls portraying gender roles which are planting in the fetus by society at large. Children learn from a very early age how they should behave, dress, speak, and what toys to play with. One must understand the term, gender, to grasp the concept of stereotyping among boys and girls. According to Margaret Mooney Marini, “Scholars now use the term sex to refer to biological based distinctions between the sexes and the term gender to refer to the social construction of differences between women and men” (Marini 95). Basically, gender means what society believes are the functions of men and women. For example; women wear pink, while a heterosexual man is called soft, and subject to having his sexuality challenged. Certain actions are associated with the female gender, such as cleaning, cooking, and taking care of children. The male gender is thought to be the one who is interested in cars and competition. This love of cars and competing against other boys is one type of gender based toy advertising. Most commercials are, either, aimed to sell to boys or girls, rarely both, (unless it is some type of board game). Many women, especially feminist, will say the girls get the short end of the diversity stick. Ayen Bakir and Kay M. Palan explain that, “One aspect that may play a significant role in how a child responds to a television advertisement is the degree to which the execution of the ad evokes his or her understanding of gender, that is, the degree to which the ad makes gender salient to the child” (Bakir, Palan 35). Children respond to what they feel is important in maintaining the fact that they are a boy or a girl. If a girl, for example, associates a beautiful pink dollhouse with being a prerequisite for being who she is, a girl, than she wants that dollhouse, because she wants to be a normal part of her society. Sense a great deal of society, especially children, receive their gender identification from outside sources, such as television, it is a win win for the advertiser. They sell their product, and keep little girls from thinking outside the box. Thus, making the job of the copywriter an easy, albeit lazy, occupation. The Barbie commercial below is an example of gender stereotyping. All of the girls are dressed in pretty, pastel colors, indicating the soft and sweetness of little girls. Most adults are wise enough to know that all little girls are not the saccharine images portrayed in this commercial. However, most commercial appear to indicate this to be so, leaving a confused bunch of soccer, and softball loving young girls. While, girls playing sports is much more common now than it used to be, some girls are still called tomboys if they play them. Being competitive is associated with the male gender. The music on the commercial even evokes certain gentleness because children (they sound like girls, but one should not assume squeaky voices only belong to girls) are singing. Naturally, the dollhouse in which they are advertising is also made up of different light shades of pink, purple, and blue. They are also inside the house, which is also a common trait with girls. The characteristics of this commercial, definitely, do not belong to the advertising of little boy products. This commercial for Monster Trucks displays a, decidedly different method for reaching its audience of boys. In accordance with gender stereotypes, this thirty second commercial is loud. Most boys, let us face it, are perceived as loud. Therefore, the narrator or voiceover person presents a deep, commanding voice. It is violent because, right from the beginning, a huge monster hand tears down a mountain of dirt. The music in this commercial resembles one of a platoon marching with a drill Sergeant. This, of course, is exactly the point. Boys associate the military with manliness, because this is what society tells them males do. It evokes power for boys and men. The advertisement, even, uses the words big and powerful throughout. However, just because boys are often seen as the powerful of the genders, it is not always a good thing. Lois J. Smith notes that, “It is easy to see the limitations for girls’ behaviors in existing television commercials, but the boys also have limitations. Ads do not portray them as nurturing or sharing. Commercial messages often show them as aggressive, physically active, and needing to win. What of the boy who could play a different role? Just as girls should not be limited to their homes, boys should be allowed to be kind and sharing” (Smith 15). Confining one’s child to the activities specific to what society says male and female genders should do is doing a disservice to both sexes. When, as Jennifer J. Pike and Nancy A. Jennings say, “Commercials present gender stereotypes through overt factors, such as activities and language, as well as through more subtle features, such as voiceovers and production features” (1195), it is hard to do away with what a child believes he/she is capable of. The ramification of these images stays in the mind of girls and boys well into their adulthood. This explains why men always tend to want to be seen as the stronger sex, and women are eager to let them believe they are. Most of society believes that women are the physically weaker sex, while men are the emotional weaker sex because of the way the sexes are tied to gender specifications. Yes, there is truth in the fact that most men are stronger; yet, some women can hold their own against a man. Perhaps, not with brute force, but with other forms of combat, such as martial arts, and other self-defense tactics. Yet, until recently, one would never believe this to be true. This type of female empowerment usually reserved for adult women to play. Strong women are portrayed in films like The Matrix franchise and various movies staring Angelina Jolie. There are also movies showing men in a more sensitive light such as American Beauty with Kevin Spacey, and Revolutionary Road with Leonardo DiCaprio. These types of films show men and women representing characteristics which are opposite of what society says their gender should represent. The scene below represents ashift towards changing the gender roles in which women and men are chained to. The first scene shows Angelina Jolie’s character of Mrs. Smith is seen kicking some serious behind in the movie, Mr. and Mrs. Smith. Not only does she shoot at the bad guys, she is a bad guy, herself. Jolie plays an assassin who can drive a car just as a man is thought to be able to do. She can handle several firearms at once. She is fearlessly, fierce, and performing all of the action movies in which most men, including Brad Pitt, himself, is able to perform. These are the kind of roles she is famous for. In contrast there are male parts which may remind people more of how society thinks women should be. Women are thought to be the more emotional of the two sexes, as far as gender functions in this society, particularly, American Society. Usually, men do not make a habit of showing their feelings and anguish. Yet, as shown below, in the movie Revolutionary Road, Leonardo DiCaprio portrays anguish to perfection. In the movie, (a decent clip was difficult to find) he portrays a man confused about his options in life. Although, he has a good job, somehow he stills feels empty. So when his wife, portrayed by the magnificent, Kate Winslet, wants to move to Paris, he, uncharacteristically of a male, agrees to pack up all of their belongings and move. Most of the time the male gender would choose to be sensible in movies and television. Usually it is the woman, who is happens to be confined to the gender role of only being a wife and a mother, wants a change. Although, in the end of the movie he decides because of a promotion, it was refreshing to see him, at least, think about another alternative of living. Yet, the movie reverts back to what a typical male, especially during the 1950s, would do. He got a promotion and changed his mind. However, when the wife finds out she is pregnant with their third child, in another reversal of gender roles, he is the one who wants to keep the baby. While, Winslet does not. The movie, although tragic in the end, presents what is seen as gender specific, and the tragedy of those stereotypes. Women may not have necessarily wanted many children, because of America’s history of oppression of women, and gender stereotypes which have taught them it is wrong to not want babies. This movie is a perfect example of what this type of discourse can occur as a result of gender blindness. Because she felt trapped in motherhood she kills herself with a botched self-inflicted abortion. While, this movie presents the worst thing which could happen because of gender biased, the reality for this kind of tragedy still exist. Yet, all is not gloom and doom for the outcome of America’s stereotypes and its affect on children, as well as adults. There are some glimpses of hope for young men and women of the future to get past the hurdles of gender based advertisements and television shows. As Sarah Banet-Weiser says, “In the past decade the representation of girls on television has been influenced by the more general mainstreaming of feminist rhetoric, and Nickelodeon has led the way in terms of children’s television” (Banet-Weiser 125). Many shows on Nickelodeon feature teenagers who are empowered, and have the same type of confidence and outspokenness, generally, associated with boys. Shows such as, I Carly, Hannah Montana (whether one likes it or not), and a few others of the same caliber express young women in a variety of ways. The character of Carly, played by teen, Miranda Cosgrove, on I Carly has her own web show. She produces her own show with the help of another female teen character, Sam, who is portrayed by Jennette McCurdy. Sam’s character truly steps out of the boundaries of gender assumptions. She is tough and does not take anyone disrespecting her. McCurdy plays her with such tenacity one does not think about her being a female character pretending to be tough, she is tough on her own merit. Usually in the media the teenage boy is thought to be the aggressor. She blows this stereotype out of the water. This montage of clips featuring McCurdy says it all. Who would not want their little girl to be able to stand up for herself in this manner? She is not waiting on the typical male, according to stereotypical gender roles, to bail her out of any situation. Sam is her own hero. Even, if one is not a feminist this kind of chutzpa from a young girl or boy is to be admired. So, hats off to the kid. Society can only hope that one day we will get past such stereotypes in the near future. The media has a way of making us believe what we are watching, listening too, and reading. It is up to the individual, and individual parents, to let their children understand that they do not ever have to be trapped by what this society tells them they should act, or be. Who knows if we will ever get there, but television stations like Nickelodeon will help the cause. One can see Jennette McCurdy as the next female action hero. And when she does become one it will not be an issue. What a day that will be. Baker, Aysen and Kay M. Paylan. “How are Children’s Attitudes Towards Ads and Brands Affected by Gender-Related Content in Advertising.” Journal of Advertising. (Spring 2010) V39 1, p35. Banet-Weiser, Sarah. “Girls Rule!! Gender, Feminism and Nickelodeon.” Critical Studies in Media Communication. (June 2004) V21,N.2. Pp 119-139. Mooney-Marini, Margaret. “Sex and Gender: What Do We Know.” Sociological Forum. (Mar. 1990) V5,1. Pp 95-120. Pike, Jennifer J. and Nancy A. Jennings. “The Effects on Commercials on Children’s Perceptions of Gender Appropriate Toy Use.” Sex Roles. (January 2005) V.52, No ½. Pp 83-91. Smith, Lois J. “A Content Analysis of Gender Differences in Children’s Advertising.” Journal of Broadcasting and Electronic Media. (Summer 1994), V.38 I 3, p323.

<urn:uuid:f44b26cd-05e7-4990-8b21-1e3e867d572b>

Courses Courses for Kids Free study material Offline Centres More Store # The quotient when $1 + {x^2} + {x^4} + {x^6} + .......... + {x^{34}}$ is divided by $1 + x + {x^2} + {x^3} + ........... + {x^{17}}$. isA. ${x^{17}} - {x^{15}} + {x^{13}} - {x^{11}} + ......x$B. ${x^{17}} + {x^{15}} + {x^{13}} + {x^{11}} + ..........x$C. ${x^{17}} + {x^{16}} + {x^{15}} + {x^{14}} + ..........x$D. ${x^{17}} - {x^{16}} + {x^{15}} - {x^{14}} + .......... + x$ Last updated date: 24th Jun 2024 Total views: 405k Views today: 4.05k Verified 405k+ views Hint: In order to solve this problem we need to use the formula of sum of finite geometric series $S = \dfrac{{a({r^n} - 1)}}{{r - 1}}$ . Then we need to use it for both the series given and then we need to divide the obtained sum to get the right answer. As we know, the sum of finite geometric series $1,r,{r^2},{r^3},{r^4}..........{r^{n - 1}}$ with common ratio is given by $a,ar,a{r^2},a{r^3},a{r^4}..........a{r^{n - 1}} = S = \dfrac{{a({r^n} - 1)}}{{r - 1}}$ = sum (where r is greater than 1) When r is less than 1 then the formula of sum for n terms of GP will be $\dfrac{{a(1 - {r^n})}}{{1 - r}}$. Now, $1 + {x^2} + {x^4} + {x^6} + .......... + {x^{34}}$ forms a finite geometric series with common ratio ${x^2}$, first term 1 and number of terms n = 18 (since power of x from 2 to 34 with the difference of 2 is 17 terms and the first term is 1 therefore n = 18) then sum is, $1 + {x^2} + {x^4} + {x^6} + .......... + {x^{34}} = \dfrac{{{x^{2(18)}} - 1}}{{{x^2} - 1}}$ $1 + {x^2} + {x^4} + {x^6} + .......... + {x^{34}} = \dfrac{{{x^{36}} - 1}}{{{x^2} - 1}}$……….(1) Now, $1 + x + {x^2} + {x^3} + ........... + {x^{17}}$ forms a finite geometric series with common ratio x, first term 1 and number of terms n = 18 (since power of x from 1 to 17 is 17 terms and first term is 1 therefore n = 18) then sum is $1 + x + {x^2} + {x^3} + ........... + {x^{17}} = \dfrac{{{x^{18}} - 1}}{{x - 1}}$……………..(2) Now, Dividing (1) by (2) equation $\Rightarrow \dfrac{{\left( {{x^{36}} - 1} \right)\left( {x - 1} \right)}}{{\left( {{x^2} - 1} \right)\left( {{x^{18}} - 1} \right)}} = \dfrac{{{x^{18}} - 1}}{{x + 1}}$ Now, we have to divide${x^{18}} + 1$ by (x + 1). We can see the degree of the numerator is 18 and the denominator has a linear equation. So, the degree of quotient starts from 17 and decreases with a difference of one. $\Rightarrow \dfrac{{{x^{18}} + 1}}{{x + 1}} = {x^{17}} - {x^{16}} + {x^{15}} - {x^{14}} + .......... + x$ So, the correct answer is “Option d”. Note: Whenever we face such types of problems we use some important points. Like first of all try to convert series into easy form by using sum of series. While dividing carefully observe the degree of quotient and sign of coefficient. Doing these things will solve all such problems.

crawl-data/CC-MAIN-2024-26/segments/1718198864968.52/warc/CC-MAIN-20240623225845-20240624015845-00860.warc.gz

The ozone hole reached its maximum size for the year Sept. 22, covering 8.2 million square miles -- the area of the United States, Canada and Mexico combined. The ozone layer acts as Earth's natural shield against ultraviolet radiation, which can cause skin cancer, and the ozone hole phenomenon began making a yearly appearance in the early 1980s. However, the size of the hole has decreased in recent years after an international agreement regulating the production of certain ozone-depleting chemicals. Warmer temperatures in the Antarctic lower stratosphere helped to keep the hole smaller this year, NASA said. "The ozone hole mainly is caused by chlorine from human-produced chemicals, and these chlorine levels are still sizable in the Antarctic stratosphere," said Paul Newman, an atmospheric scientist at NASA's Goddard Space Flight Center in Greenbelt, Md. "Natural fluctuations in weather patterns resulted in warmer stratospheric temperatures this year. These temperatures led to a smaller ozone hole." NASA and the National Oceanographic and Atmospheric Administration have been monitoring the ozone layer from the ground and with a variety of instruments on satellites and balloons since the 1970s. The Antarctic ozone layer likely will not return to its early pre-1980s state until about 2065, Newman said, because of the long lifetimes of ozone-depleting substances still in the atmosphere.

<urn:uuid:c13075fa-07f3-4a82-90d2-87432b2659ac>

Homework Help: Basic Proof Writing Help continued. 1. May 30, 2015 Keen94 1. The problem statement, all variables and given/known data Given a series of mathematical statements, some of which are true and some of which are false. Prove ∀x: 3x2-5x-2=(3x+1)(x-2) 2. Relevant equations x(px→qx) 3. The attempt at a solution Statement: ∀x: 3x2-5x-2=(3x+1)(x-2) x{x∈ℝ I 3x2-5x-2=(3x+1)(x-2)} (1) Assume 3x2-5x-2 is true [Hypothesis] (2) 3x2-5x-2 has two real roots. [Some Definition "?"] (3) (3x+1)(x-2) =3x2-5x-2 =(3x+1)(x-2) Therefore 3x2-5x-2=(3x+1)(x-2) for all x by definition "?". What definition would I be using? Is this the right way to go about this proof? Should I be cataloging definitions and previously proved theorems? 2. May 30, 2015 Staff: Mentor No. 3x2 - 5x - 2 is an expression, not a statement. Expressions have values, but "true" and "false" are not possible values. A statement, such as an equation or inequality or a sentence such as "I have three cats." can be true or false. The hypothesis here is "x is a real number." The conclusion is $3x^2 - 5x - 2 = (3x + 1)(x - 2)$. The = symbol here is really $\equiv$, since you are showing that the statement is identically true (for all real x), not just true for a small handful of value of x. Technically, no. The equation $3x^2 - 5x - 2 = 0$ has two real roots, but I think you're heading down the wrong track here. It seems to me that one could prove this by expanding (i.e., multiplying out) the right side. The above looks to me like you are proving that (3x + 1)(x - 2) is equal to itself, which is certainly true, but a trivial result. 3. May 30, 2015 Keen94 Hmmm... I can see this is a tautology involving two expressions. Does this really boil down to showing the LHS is equal to the RHS as you mentioned? Because I don't see any other way to do this since it is an equivalence between two expressions. 4. May 30, 2015 Keen94 ***update*** x: 3x2-5x-2=(3x+1)(x-2) If x is a real number, then for all x, 3x2-5x-2=(3x+1)(x-2) (1) Assume U=ℝ (2) Let x∈ℝ (3) 3x2-5x-2=(3x+1)(x-2) =3x2-6x+x-2 =3x2-5x-2 Therefore 3x2-5x-2=(3x+1)(x-2) for all x, where x∈ℝ. 5. May 30, 2015 Keen94 In contrast to the following. The sum of the roots of: x2+5x+4=0 is equal to 5. (1) Let U=ℝ and x∈ℝ. Assume ∃x{x I x2+5x+4=0} (2) x2+5x+4=0 has real roots, a and b, for which a+b=5. (3) 0=x2+5x+4 0=(a+4)(b+1) 0=a+4 or 0=b+1 a=-4 or b=-1 a+b=5 (-4)+(-1)=5 -5≠5 Therefore it is false that the sum of the roots of x2+5x+4=0 is 5. 6. May 30, 2015 HallsofIvy I would think that the simplest proof that $x^2+ 5x+ 4= (x- 4)(x- 1)$ would be to start from $(x- 4)(x- 1)$ and show that it results in $x^2+ 5x+ 4$. Start with $(x- 4)(x- 1)= x(x- 1)- 4(x- 1)$ (distributive law) and continue from there. 7. May 30, 2015 Staff: Mentor No, the variable is x. Your conclusion is correct, but you are trying to make the work you show fit into a "one size fits all" pattern. The statement here is that the roots of the given quadratic equation add up to 5. All you need to do is to find the two roots, using factorization (which you did, sort of) or the quadratic formula. Unless your teacher is unusually pedantic, all of this "Let U=ℝ and x∈ℝ. Assume ∃x{x I x2+5x+4=0}" is completely unnecessary. Statement: The sum of the roots of: x2+5x+4=0 is equal to 5. Find roots: $x^2 + 5x + 4 = 0$ $\Leftrightarrow (x + 4)(x + 1) = 0$ $\Leftrightarrow x = -4 \text{ or } x = -1$ The two roots of the equation add to -5, so we conclude that the statement is false. 8. May 30, 2015 Keen94 wow, so much simpler. Thanks again for the help guys.

crawl-data/CC-MAIN-2018-22/segments/1526794867254.84/warc/CC-MAIN-20180525235049-20180526015049-00029.warc.gz

Under the Microscope By Brendan I. Koerner Why Judges Rarely Change Their Minds By Edward Lazarus Net Loss By Reed E. Hundt History Lesson By Jack M. Balkin Five Supreme Court justices think Congress doesn't have the power to pass new laws against discrimination. They're forgetting about the civil rights movements of the 19th and 20th centuries. The civil rights act of 1964 is one of the great achievements of American law and, together with the Voting Rights Act of 1965, the crowning accomplishment of the civil rights movement of the 1950s and 1960s. The 1964 law prohibits discrimination on the basis of race, sex, national origin, or religion, at work and in schools, restaurants, businesses, and other establishments that are open to the public. It is the model for almost every civil rights law that has followed and has probably done more to guarantee equal opportunity for Americans than any United States Supreme Court decision, including the ban on school segregation in the historic case Brown v. Board of Education. You would think that so basic a symbol of American liberty and equality must be within Congress's power to enact. Yet when Congress debated the act, the source of that power was by no means clear. Even today, Congress's authority to pass civil rights laws remains a profound problem of constitutional law. The Constitution's Fourteenth Amendment, ratified in 1868, fundamentally altered the balance of state and federal power. It prevented states from denying basic civil rights and gave Congress the power to enforce its guarantees of liberty and equality. But in the decades following Reconstruction, the Supreme Court became hostile to the rights of blacks and wary of Congressional interference in states' affairs. The Court drastically limited Congress's civil rights power by narrowly interpreting the Fourteenth Amendment, striking down many Reconstruction-era civil rights laws, and looking the other way as Southern state governments systematically oppressed blacks. Generations later, in the wake of the civil rights movement, the Supreme Court, led by Chief Justice Earl Warren, upheld the 1964 Civil Rights Act as constitutionalbut not because of Congress's power to pass laws under the Fourteenth Amendment. Instead, the court turned to the Constitution's Commerce Clause, which gives Congress the power to regulate interstate commerce. Rather than treating the evils of discrimination as an affront to justice and equalityand revisiting decisions it had made 80 years earlierthe court recast those evils as a barrier to the free flow of commerce. Picking up on this legal fiction, Congress has repeatedly invoked the Commerce Clause as the source of its power to pass laws broadening equality of opportunity and protecting workers from discrimination based on pregnancy, age, or disability. The clause has become the key vehicle both for regulating the American economy and for defending civil rights. Now, however, a five-person conservative majority on the Supreme Court has begun to take down the Warren Court's legal construction. Chief Justice William Rehnquist, joined by Associate Justices Antonin Scalia, Clarence Thomas, Sandra Day O'Connor, and Anthony Kennedy, has embarked on a crusade to impose new constitutional limits on federal power, and the most important targets have been civil rights laws. In three recent major decisions, the court has curbed Congress's power to pass new civil rights legislation under both the Commerce Clause and the Fourteenth Amendment. In the name of states' rights, the court struck down a federal law that sought to remedy violence against women in United States v. Morrison, and it held that state employees cannot sue for damages when they suffer discrimination based on age or disability in Kimel v. Florida Board of Regents and in Board of Trustees of the University of Alabama v. Garrett. These decisions are part of a revolution in constitutional law that runs counter to the country's deepest commitments to equality and liberty. Morrison, Kimel, and Garrett are bad law because they throw roadblocks in the way of future civil rights protection. But they're also bad history, based on a kind of amnesia: They were written by justices who seem to have forgotten the meaning of the nation's many struggles for equal citizenship, from Reconstruction to the fight for women's suffrage to the civil rights movement. When the framers of the constitution were first debating it, few people imagined that Congress would prove to be the basic guarantor of American liberties. In fact, the Bill of Rights was added in 1791 to limit Congress's power and prevent the new federal government from interfering in local affairs. Many members of the founding generation believed that a weaker central government would help secure republican liberty, by which they meant not simply individual freedom but also the preservation of the existing social order within the statesan order that included chattel slavery and limited the vote to white male property owners. Although Kentucky and Virginia protested vigorously against a 1798 federal law restricting political speech, for example, they raised no objection to state-based speech restrictions. The Civil War changed all this. Given that several states had held blacks in slavery for generations, the states no longer seemed like the primary guarantors of liberty. Even after the Thirteenth Amendment ended slavery in 1865, Southern state governments denied blacks basic civil rights through the infamous Black Codes, which reduced blacks' economic opportunities to being sharecroppers and servants for whites and severely punished blacks for trying to leave their white employers. The framers of the Fourteenth Amendmentthe so-called Radical Republicansconcluded that the states had defaulted as protectors of civil liberties. The Fourteenth Amendment reflected very different assumptions from those of the founding generation. From now on, Congress rather than the states would be a central protector of basic rights, which the amendment called "the privileges or immunities of citizens of the United States." No state could abridge those privileges or immunities, or deny any person due process or the equal protection of the law. And to make sure that states did not violate people's constitutional rights, Congress gave itself the power to enforce the amendment through "appropriate legislation." The Reconstruction Congress quickly used this new power. The Enforcement Acts of 1870 and 1871 banned state laws that denied blacks the right to vote, outlawed fraudulent voter-registration practices, and authorized federal court supervision of suspicious elections. The Ku Klux Klan Act of 1871 prevented illegal intimidation of blacks where states were unwilling or unable to provide protection, making it a federal crime for private parties to conspire to violate civil rights. And the Civil Rights Act of 1875 promised full and equal access to "inns, public conveyances on land or water, theaters, and other places of public amusement," anticipating by almost a century several of the provisions of the 1964 Civil Rights Act. The justification for this expanded federal power was genuinely new: For the first time, civil rights became a matter of national concern. Here at last, it seemed, was the source of Congressional power to pass federal civil rights legislation. But reactionary forces challenged that view. Opponents of ReconstructionNorthern Democrats and conservative Republicansdisagreed with the Radical Republicans about the meaning of the Civil War, and hence about the scope of the new federal power created by the Fourteenth Amendment. To them, the Civil War was about the illegality of secession, not the injustices of slavery or the failure of state governments to protect civil rights. Once slavery was abolished, and the Southern states welcomed back into the Union, there was no need for federal interference in the states' internal operations. The Supreme Court adopted the Northern Democrat view in the 1873 Slaughterhouse Cases, as the sociologist Pamela Brandwein of the University of Texas at Dallas has explained. Worried that the Fourteenth Amendment's "privileges or immunities" clause would give the federal government enormous new power to supervise states, Justice Samuel Miller effectively wrote the clause out of existence. When voting irregularities left the presidential election of 1876 in dispute, white Northern and Southern politicians resolved the controversy with the Compromise of 1877, in which the federal government withdrew its troops from the South and ended Reconstruction. The compromise handed Southern state governments back to white "Redeemers," who were determined to restore white supremacy through public power and private terror. Redeemer governments used endless subterfuges to deny blacks equal opportunities while turning a blind eye to lynchings, intimidation, and violence. According to a new conventional wisdom, Reconstruction's attempt to foist racial equality on whites had gone too far, and the Civil War had not fundamentally changed the relationship between the states and the federal government. The new racial and political conservatism soon dominated the Supreme Court. In the name of states' rights, the court began to strike down the civil rights laws passed by the same Congress that had written the Fourteenth Amendment. In 1883, in United States v. Harris, the court declared the criminal provisions of the Ku Klux Klan Act of 1871 unconstitutional, refusing to punish a white lynch mob in Tennessee on the grounds that Congress could not reach private conspiracies through its Fourteenth Amendment power. The same year, in the egregiously misnamed Civil Rights Cases, the court struck down the last great achievement of the Reconstruction Congress, the Civil Rights Act of 1875. Justice Joseph P. Bradley argued that allowing Congress to protect blacks from private discrimination undermined state sovereignty and insulted states by assuming that they would not protect their black citizens. He dismissed the Civil Rights Act for making blacks "the special favorite of the laws." That vision informs much of the court's jurisprudence from the 1880s onward. In the infamous 1896 decision Plessy v. Ferguson, which upheld Louisiana's segregation of railway carriages, the Supreme Court gave its blessing to Jim Crow by creating the doctrine of "separate but equal" facilities that would be used to justify segregation for decades. In 1903 in Giles v. Harris, the Supreme Court simply looked the other way when Alabama disenfranchised its black citizenry, declaring that there was nothing it could do even if federal constitutional rights were openly violated. Other Southern states quickly got the message, and by World War I, the South was rigidly segregated and blacks were shut out of the political process. Few people today realize how important the Civil Rights Cases were. As Yale law professor Akhil Reed Amar has pointed out, if the Supreme Court had upheld Congress's power to protect civil rights in 1883, the 1875 Civil Rights Act would have trumped the state law that segregated railway carriages in Plessy, as well as much of the Jim Crow legislation that swept the South in the early 20th century. By invoking the shibboleth of states' rights to limit Congressional power, the Supreme Court helped crush equal opportunity for blacks for generations. Ninety years after the ill-fated Civil Rights Act of 1875, the modern civil rights movement pressed for a bill to protect blacks from discrimination in housing, jobs, and public places. A great deal had changed in the interim, including the meaning of the term "civil rights." Justice Bradley's view, widely shared in the late 19th century, was that people had civil rights only in relation to dealings with the government. If the government tried to restrict your right to make contracts or own property because of your race, or failed to enforce laws to protect you, you could complain that your civil rights were violated. But a private person could not violate them by definition. To the leaders of the 1960s civil rights movement, this view was bizarre. For them the most important abridgments of civil rights involved private acts of discriminationby employers who refused to hire blacks or restaurant owners who refused to serve them at lunch counters. Many of the movement's standard methods of protest, like boycotts and sit-ins, were aimed at private businesses, not the state. In promoting the new civil rights bill, the Kennedy and Johnson administrations faced a quandary: The Civil Rights Cases and Harris seemed to foreclose using Congress's power to enforce the Fourteenth Amendment and prevent private discrimination, and there was no guarantee that the Supreme Court would overturn those 80-year-old precedents. So the administrations offered an additional theory of Congressional power. A generation earlier, the constitutional struggle over the New Deal had established that the federal government had broad authority to regulate the national economy. Through its power under the Commerce Clause, Congress could enact laws about anything that used or traveled through the instruments of interstate commercehighways, trains, telephones, mailand it could regulate anything that might substantially affect interstate commerce. In 1942, the Supreme Court held that Congress could even regulate wheat grown on a family farm for home consumption because, it reasoned, the cumulative effects of home-grown produce could affect the national market. After that, most constitutional scholars assumed that the federal government could regulate just about anything through this power. Thus, in December 1964, the Warren Court upheld the new Civil Rights Act as a valid regulation of interstate commerce. Congress, the court ruled, could reasonably conclude that segregated restaurants and hotels used food shipped through interstate highways or railways, and that segregation discouraged blacks from spending money and traveling between states. The Warren Court has a reputation for bucking the will of majorities. But as University of Texas law professor Lucas A. Powe Jr. has persuasively argued, the court was remarkably deferential to Congress and promoted the values of national majorities over regional majorities, particularly those in the South. Far from viewing the new Civil Rights Act with suspicion, Warren and his colleagues were eager to uphold it. For months before the act passed, sit-in cases had been percolating up to the Supreme Court. Lower courts throughout the South were convicting protesters for trespassing on the property of segregated white businesses. Anxious not to derail the civil rights movement, the court had reversed many convictions on technicalities, but thousands of cases were still active. Soon the court would have to decide whether using state trespass laws to keep black protesters out of segregated facilities violated the provision of the Fourteenth Amendment that no state could deny equal protection of the laws. Yet even if the court ordered stores and restaurants to serve blacks, there was no guarantee that Southern businesses would comply. The 1964 Civil Rights Act took political pressure off the court by putting the authority of a democratically elected Congress behind the civil rights movement. As Robert C. Post of Boalt Hall at the University of California, Berkeley, and Reva B. Siegel of Yale Law School have pointed out, the new Civil Rights Act made Congress the court's partner in articulating guarantees of equality. The 1964 act let the court interpret the Fourteenth Amendment more narrowly, secure in the belief that Congress could address private discrimination through the new act and subsequent laws. Thus, the new civil rights law that arrived on the court's doorstepand the Commerce Clause theory of Congressional powerwas a godsend. The court didn't have to overturn the 80-year-old precedents of the Civil Rights Cases and United States v. Harris. Instead, the Warren Court treated Congress's commerce power as a civil rights power: Congress had the power to keep the channels of interstate commerce clear of racial discrimination. Without directly confronting and overturning the racist Civil Rights Cases, the Warren Court effectively performed an end-run around them. All Congress had to do to pass civil rights legislation was to distort reality, by arguing that denying people their civil rights would harm commerceinstead of asserting that it violated a constitutional guarantee of equal citizenship. In succeeding years, Congress used its commerce power to pass laws that banned discrimination based on race, sex, religion, pregnancy, age, and disability, in areas ranging from education to housing to employment. As a result of the civil rights movement and Congress's response to it, it began to seem obvious to most Americans that equality, like the economy, was a subject of national concern. The Warren Court's solution was clever, but it was a lawyer's trick and a stopgap measure. The ploy worked only as long as everyone recognized that the lesson of the civil rights movement was that Congress could protect the civil rights of Americans against public and private discrimination, and that the old constitutional vision of the Compromise of 1877 had been rejected if not explicitly overruled. As soon as people forgot that lesson, or refused to acknowledge it, Congress's power to pass new civil rights laws would be on shaky ground. Chief Justice William Rehnquist is the leader of the conservative majority on today's Supreme Court. Throughout his career on the court, which began in 1972, Rehnquist has been deeply skeptical of liberal civil rights claims, and he has long pushed for a "federalist" shift of power from Congress back to the states. When Clarence Thomas replaced Thurgood Marshall in 1991, few people understood that Thomas's most important role would be as the crucial fifth vote for a revolution in federal-state relations. In the decade since his appointment, Thomas, Rehnquist, Scalia, Kennedy, and O'Connor have struck down federal law after federal law in the name of states' rights. Among the most important casualties of this new conservative judicial activism have been civil rights laws. In 1994, after years of hearings, Congress passed the Violence Against Women Act, which created a civil rights remedy that gives women who have been assaulted because of their gender a right to sue their attackers in federal court. During the course of drafting and debating VAWA, Congress found that many state criminal and civil justice systems failed to take violence against women as seriously as they did other violent crimes. Attorneys general from 38 states urged passage of the act, arguing that "the current system for dealing with violence against women is inadequate." In 2000, however, the five conservative justices struck down VAWA's civil rights remedy in United States v. Morrison. Chief Justice Rehnquist ruled that Congress lacked the authority to pass VAWA under the Commerce Clause. The New Deal, he argued, had been about regulating the economy, and violence against women was not an economic activity, no matter how great an economic effect it might have. He brushed aside Congress's findings that gender-motivated violence deters women from traveling from state to state, discourages them from taking jobs or doing business across state lines, diminishes national productivity, and increases medical costs. For Rehnquist, the cumulative impact of inherently "noneconomic" activities could not justify federal lawmaking. Otherwise, he reasoned, Congress could regulate all violent crime, as well as family law, "since the aggregate effect of marriage, divorce, and childrearing on the national economy is undoubtedly significant." Nor did the Fourteenth Amendment give Congress the power to enact VAWA. In Morrison, a woman named Christy Brzonkala sued two men she said had raped her in a college dorm room. She argued that Congress could pass VAWA to enforce the Fourteenth Amendment because states had failed to provide women equal protection from violent assaults. But Rehnquist argued that even if VAWA made up for the failures of public officials, it was aimed at private actors. Pointing to the Civil Rights Cases and United States v. Harris, Rehnquist claimed that the act was beyond Congress's authority. That same year, in Kimel v. Florida Board of Regents, the five justices held that Congress could not hold states liable for damages when they discriminated against elderly employees. The following year, in Board of Trustees of the University of Alabama v. Garrett, the same group held that states could not be sued for damages when they discriminated against disabled workers. Because employment indisputably is an economic activity, Congress could use its commerce power to pass the underlying laws. But the court argued that the Commerce Clause did not give Congress the authority to undermine the sovereignty and dignity of states by forcing state governments to pay damages when they violated workers' federal rights. The principle of state sovereigntywhich the court said was guaranteed by the Constitution's Eleventh Amendmenttrumped the federal power to regulate the economy, at least with respect to the remedies that Congress could provide. The court recognized that legislation enacted under the Fourteenth Amendment could override state sovereigntybut it argued that the amendment didn't give Congress the power to pass laws addressing age or disability discrimination. The court offered a complicated argument to support this last claim. Generally speaking, when the government makes distinctions based on race or gender, the court scrutinizes these decisions closely. But when the government discriminates based on other criteria, like age or disability, the court asks only whether the decision was wholly arbitrary or irrational. In Kimel and Garrett, the majority argued that states might rationally decide to discriminate against their older and disabled employees to save money, for example. In effect, it held that federal laws that bar discrimination based on age and disability don't really enforce civil rights guaranteed by the Fourteenth Amendment. These laws are just regulations of interstate commerce, so states can't be forced to pay damages when they violate them. Kimel and Garrett have this curious effect: The court insists that the Age Discrimination in Employment Act and the Americans with Disabilities Act are fully constitutional statutes and create federal rights that fully bind state governments. But if a citizen proves that a state fired her because of her age or her disability, she can't make the state pay her for that loss, even though it has clearly violated federal law. Many things are controversial about Morrison, Kimel, and Garrett, but perhaps the most important is their reinterpretation of the nation's history. Take Morrison's ringing endorsement of the Civil Rights Cases and United States v. Harris. The authority of these decisions, Rehnquist insisted, "stems not only from the length of time they have been on the books, but also from the insight attributable to the Members of the Court at that time," because each justice "had intimate knowledge and familiarity with the events surrounding the [Fourteenth] Amendment's adoption." What Rehnquist didn't mention was that by the 1880s, the justices reflected not the vision of the Radical Republicans who wrote the Fourteenth Amendment, but the Compromise of 1877, which sold out blacks. Next, consider Morrison's claim that the federal government should stay out of family law because that's traditionally a local concern left to the states. That version of history ignores a century or more of federal regulation of the family through welfare policies, pension laws, and criminal penalties to enforce child support judgments, as law professors Judith Resnik of Yale and Jill Elaine Hasday of the University of Chicago have pointed out. In fact, Reva B. Siegel has shown that the "families are local" argument is a direct descendant of states' rights arguments used to oppose equal rights for women, in particular the right to vote. For most of U.S. history, the states treated women as dependent members of a household ruled by their husbands or fathers, with virtually no contract or property rights of their own. Siegel explains that the "tradition of federal deference to state law grew up at least in part to preserve the status order that state law enforced." Respecting local authority in family matters meant respecting men's rights to control women through the guise of protecting them. That's why the 80-year-long struggle for women's suffrage that produced the Nineteenth Amendment was continually opposed by states' rights advocates. Opponents of women's suffrage argued that a federal right to vote would undermine "local self-government" by disrupting family harmony and opening the door to federal interference in the privacy of men's homes. In fact, when women finally won the right to vote in 1920, opponents were so afraid that women's equality would weaken local authority that they tried unsuccessfully to get the Supreme Court to declare the Nineteenth Amendment unconstitutional because it violated states' rights. Today, Siegel argues, we should read the Fourteenth Amendment's guarantee of equalityand Congress's power to enforce itin light of the long struggle for women's suffrage and its success in defeating the "local authority" proponents. When Congress passes a law like VAWA, which is designed to secure equal citizenship for women, the last thing the Supreme Court should do is strike it down for interfering with traditional local control over domestic relations. As our nation's ideas about equal citizenship change, so too must our notions about federal power to protect civil rights. Probably the most curious feature of the court's version of American history is its omission of the civil rights movement itself. Reading Morrison, Kimel, and Garrett, one would think that the movement had no impact on Americans' understanding of their Constitution. But a reasonable interpretation of the civil rights movement is that civil rights and civil equality are distinctively federal concerns. The struggle over the Civil Rights Act of 1964 established that the federal government has full authority to pass laws to secure rights of equal citizenship for all Americans. The Rehnquist majority rejects this view: To these justices, the Civil Rights Act of 1964 is just another piece of economic legislation. And even though a central target of the civil rights movement was private discrimination, the Rehnquist Court insists that Congress has no inherent civil rights power to reach private conduct. According to the conservative majority, the struggles of the 1960s left untouched the narrow, racist views of the 1880s. Also striking is the Rehnquist Court's unwillingness to accept Congress as a partner. The Warren Court happily let Congress protect civil rights more broadly than the court itself would, reasoning that lawmakers could better make fine-tuned judgments. Equally important, by letting Congress take the lead in identifying which civil rights protections were necessary, the Warren Court could learn from social movements and take into account evolving popular understandings of equality. The Civil Rights Act of 1964 addressed women's rights, for example, well before the court did in the 1970s. The Rehnquist Court also rejects this approach, saying in effect that it's irrelevant that a popular consensus has grown in favor of civil rights for the elderly or the disabled, or that a democratically elected body like Congress has responded to a changing social climate. Instead, the court must guard its status as the last (and only) authority on the meaning of the Constitution. To the extent that the evolving views of Congress or the public move beyond the court's own, they must be clipped back. The Rehnquist majority insists that its version of federalism preserves individual liberty. The idea is that less federal regulation frees people from overweening federal bureaucrats and gives the states more room to experiment. But the argument falls apart when you look at how the doctrines work in practice. Because Morrison reaffirms the New Deal principle that the power of Congress can reach any "economic" behavior and any activity that makes use of the Internet, phones, mail, railways, or highways, the majority doesn't protect much from potential federal interference. And what it does protect isn't worth defending: Morrison, for example, protects the freedom to assault women. Some states criminally prosecute rape within marriage only in limited circumstances, or punish it less severely than other rape. So Morrison's holding that Congress can't legislate to protect women helps safeguard the liberty in those states of (mostly) husbands to rape their wives. In addition, because many hate crimes don't involve economic activity, they may now be beyond federal control. So the court's new federalism doctrines also help preserve the liberty of people to assault or kill others because of their race, religion, or ethnicity. This is the sort of liberty that gives federalism a bad name. The biggest problem with the Rehnquist majority's federalism is the liberties that it overlooks. Protecting women from assault and racial and ethnic minorities from hate crimes helps preserve their liberty, not to mention their equality. This is the great lesson of the Civil War and Reconstruction: To be genuinely free, blacks needed not only an end to slavery, but also equal protection from assaults and lynchings. The Ku Klux Klan Act may have limited the liberty of racist vigilantes, but it gave blacks greater freedom from fear. The Rehnquist majority doesn't seem genuinely interested in using federalism to free people from federal regulationto do that, it would have to dismantle much more of the constitutional edifice of the New Deal. Rather, it seems primarily interested in making a symbolic gesture about limited federal power in order to rein in federal civil rights laws. The majority's rhetoric displays an almost mindless faith that striking any blow against federal power necessarily makes citizens freer. But how do the Kimel and Garrett rulings increase individual liberty by holding that states don't have to pay damages when they discriminate? The only liberty these cases protect is the liberty of states to violate people's federal civil rights. How can Americans free themselves from the doctrinal mess that the Rehnquist Court has created? The best solution is the simplest and the one most rooted in American history. The Civil War, Reconstruction, and the movements for women's suffrage and civil rights have made clear that safeguarding civil rights is central to the work of the national government. Congress's power to pass civil rights laws should come, as the Reconstruction Congress intended, from the Fourteenth Amendment. It is long past time we recognized that Congress has full authority to enact legislation guaranteeing equal citizenship. When Congress passes civil rights laws, courts should not have to pretend that they are economic regulations. And when Congress applies these laws to the states, states should have to abide by them. The opening line of the Fourteenth Amendment, the Citizenship Clause, proclaims that "All persons born or naturalized in the United States, and subject to the jurisdiction thereof, are citizens of the United States and of the State wherein they reside." The Citizenship Clause was designed to overrule the Supreme Court's 1857 Dred Scott decision, which held that blacks could not be citizens and "had no rights which the white man was bound to respect." The clause establishes a principle of equal citizenship, prohibiting the idea of first- and second-class citizens. Congress's power to enforce the Fourteenth Amendment includes the power to enforce this clauseto pass all legislation that it reasonably believes promotes the goals of equal citizenship. The Citizenship Clause, like the Thirteenth Amendment's ban on slavery, says nothing about state action, and so applies to private actors as well. The civil rights movement taught us that private discrimination in buses and coffee shops can do as much damage as Jim Crow laws, and there should be no doubt that Congress has the authority to prevent both forms of discrimination. Nor is Congressional power under the Citizenship Clause limited to redressing discrimination based on race or gender. Just as the reach of Congress's commerce power has grown over the years in response to our developing economy, the reach of its civil rights power should grow as our nation comes to terms with different kinds of prejudice and inequality. Perhaps more important, when Congress promotes equal citizenship, it should not be limited to enforcing rights that judges have already recognized. Lawmakers may decide for themselves which laws are most needed, just as they may decide how best to promote the free flow of commerce. Understanding what it means to be a free and equal citizen in a democracy is an ongoing project, in which legislatures and popular understandings have as much of a role to play as do courts. Congress gave itself the power to enforce the Fourteenth Amendment because it understood that courts are not always the most enlightened actors in the American political system. The Supreme Court may fall victim to hubris, as it did in the Dred Scott case. It may fail to listen to the demands of social movements pressing for equality. And it may take too limited a view of the rights that Americans possess and fail to protect those rights that are needed most. When Congress passes a law that it claims will promote equal citizenship, courts should ask only whether that conclusion is reasonable. Put to this test Morrison, Kimel, and Garrett become easy cases: Protecting women from violence clearly helps guarantee their equal citizenship; so does allowing state employees to win damages when they prove discrimination based on age or disability. At bottom, the dispute over Congress's power to protect civil rights is a dispute about the power to protect civil rights is a dispute about the lessons of American history and about the values to which Americans are committed. We must decide, in short, if we want to be the country of the Civil Rights Cases or the country of the civil rights movement. If we remember who we are and where we have come from, that question should not be difficult to answer.

<urn:uuid:8222db8f-6f8f-43ec-b5d4-6565fcf1d407>

### Home > ACC7 > Chapter 5 Unit 3 > Lesson CC3: 5.1.2 > Problem5-22 5-22. Solve each of the following equations. Be sure to show your work carefully and check your answers. 1. $2(3x−4)=22$ Distribute, then simplify. $6x−8=22$ $6x=30$ $x=5$ 1. $6(2x−5)=−(x+4)$ Begin this problem by distributing the 6 and the negative that are outside of the parentheses, then simplify by isolating the variable onto one side of the equation. $12x−30=−x−4$ $13x−30=−4$ $13x=26$ $x=2$ 1. $2−(y+2)=3y$ Use the same methods outlined in parts (a) and (b) to solve parts (c) and (d). Check your answer. 1. $3+4(x+1)=159$ $x=38$

crawl-data/CC-MAIN-2024-38/segments/1725700652130.6/warc/CC-MAIN-20240920022257-20240920052257-00187.warc.gz

# Calculus Applied to Probability and Statistics by Stefan Waner and Steven R. Costenoble ## This Section: 4. You're the Expert Creating a Family Trust 2. Probability Density Functions: Uniform, Exponential, Normal, and Beta 3. Mean, Median, Variance and Standard Deviation Calculus and Probability Main Page "Real World" Page 4. You're the Expert Creating a Family Trust Your position as financial consultant to the clients of Family Bank, Inc., often entails your having to give financial advice to clients with complex questions about savings. One of your newer clients, Malcolm Adams, recently graduated from college and 22 years old, presents you with a perplexing question. "I would like to set up my own insurance policy by opening a trust account into which I can make monthly payments starting now, so that upon my death or my ninety-fifth birthday-whichever comes sooner-the trust can be expected to be worth \$\$500,000.\$ How much should I invest each month?" This is not one of those questions that you can answer by consulting a table, so you promise Malcolm an answer by the next day and begin to work on the problem. After a little thought, you realize that the question is one about expected value-the expected future value of an annuity into which monthly payments are made. Since the annuity would terminate upon his death (or his ninety-fifth birthday), you decide that you need a model for the probability distribution of the lifespan of a male in the United States. To obtain this information, you consult mortality tables and come up with the histogram in shown in the following figure (you work with the actual numbers, but they are not important for the discussion to follow). From the data you calculate that the mean is \$µ = 70.778\$ and the standard deviation is \$σ = 16.5119.\$ Next, you decide to model these data with a suitable probability density function. You rule out the uniform and exponential density functions, since they have the wrong shape, and you first try the normal distribution. The normal distribution is \$f(x) =\$ Using the above values for \$µ\$ and \$σ,\$ you obtain the following figure, which shows the graph of the normal density function superimposed on the actual data. This does not seem like a very good fit at all! Since the actual histogram looks as though it is "pushed over" to the right, you think of the beta distribution, which has that general shape. The beta distribution is given by \$f(x) = (β+1)(β+2)x^β(1-x),\$ where \$β\$ can be obtained from the mean \$µ\$ using the equation \$µ = (β + 1)/(β + 3).\$ There is one catch: the beta distribution assumes that \$X\$ is between \$0\$ and \$1,\$ whereas your distribution is between \$0\$ and \$100.\$ This doesn't deter you: all you need to do is scale the \$X\$-values to \$1/100 = 0.01\$ of their original value. Thus you substitute \$µ = (0.01)(70.776) = 0.70776\$ in the above equation and solve for \$β,\$ you obtain \$β = 3.8437.\$ You then plot the associated beta function (after scaling it to fit the range \$0  ≤  X  ≤  100)\$ and again discover that, although better, the fit still leaves something to be desired. Now you just want some function that fits the data. You turn to your statistical software and ask it to find the cubic equation that best fits the data using the least squares method. It promptly tells you that the cubic function that best fits the data is \$f(x) = ax^3 + bx^2 + cx + d,\$ where \$a = -3.815484×10^{-7}\$ \$b = 5.7399145 × 10^{-5}\$ \$c = -0.0020856085\$ \$d = 0.0190315095.\$ Its graph is shown below. Note that the curve, although erratic for small values of \$X,\$ fits the large peak on the right more closely than the others. Encouraged, you use the same software to obtain a quartic (degree 4) approximation, and you find: \$f(x) = ax^4 + bx^3 + cx^2 + dx + e,\$ where \$a = -9.583507×10^{-9},\$ \$b = 1.650155×10^{-6},\$ \$c = -8.523081×10^{-5},\$ \$d = 0.0016190575\$ \$e = -0.007865381.\$ The following figure shows the result. This seems like the best fit of them all-especially for the range of \$X\$ you are interested in: \$22  ≤  X  ≤  95.\$ (Malcolm is \$22\$ years old and the trust will mature at \$95.\$) Now that you have the density function you wish to use, you use it to find the expected future value of an annuity into which monthly payments are made. The simplest formula for the future value \$V\$ of an annuity is \$V = 12P[ (1 + i/12)^{12n} - 1 ]/i.\$ (This is a standard formula from finance. This formula assumes that interest is paid at the end of each month.) Here, \$P\$ is the monthly payment-the quantity that Malcolm wants to know-\$i\$ is the interest rate, and \$n\$ is the number of years for which payments are made. Since Malcolm will be making investments starting at age \$22,\$ this means that \$n = x-22,\$ so the future value of his annuity at age \$x\$ is \$V(x) = 12P[ (1 + i/12)^{12(x-22)} - 1 ]/i.\$ As for the interest rate \$i,\$ you decide to use a conservative estimate of \$5%.\$ Now the expected value of \$V(X)\$ is given by \$E(V) =\$ ∫2295 \$V(x)f(x) dx.\$ where \$f(x)\$ is the quartic approximation to the distribution function. Since Malcolm wants this to be \$\$500,000,\$ you set \$500,000 =\$ ∫2295 \$V(x)f(x) dx.\$ \$=\$ ∫2295\$12P[ (1 + i/12)^{12(x-22)} - 1 ]f(x)/i  dx\$ \$= P\$ ∫2295\$12[ (1 + i/12)^{12(x-22)} - 1 ]f(x)/i  dx.\$ Solving for \$P,\$ \$P = 500,000 / {\$ ∫2295\$12[ (1 + i/12)^{12(x-22)} - 1 ]f(x)/i  dx }\$ You now calculate the integral numerically (using the quartic approximation to \$f(x)),\$ obtaining \$P\$ à\$ 500,000/3,409.8019 ≈  146.64\$ per month. The next day, you can tell Malcolm that at a \$5%\$ interest rate, his family can expect the trust to be worth \$\$500,000\$ upon maturity if he deposits \$\$146.64\$ each month. Exercises 1. How much smaller will the payments be if the interest rate is \$6%\$? 2. How much larger would the payments be if Malcolm began payments at the age of 30? 3. Repeat the original calculation using the normal distribution described above. Give reasons for the discrepancy between the answers, explaining why your answer is smaller or larger than the one calculated above. 4. Repeat Exercise 3 using the cubic distribution. 5. If Malcolm wanted to terminate the trust at age \$65,\$ which model would you use for the probability density? Give reasons for your choice. 6. Suppose you were told by your superior that, since the expected male lifespan is \$71,\$ you could have saved yourself a lot of trouble by using the formula for the future value of an annuity maturing at age \$71.\$ Based on that, the payment comes out to about \$\$198\$ per month. Why is it higher? Why is it the wrong amount? 7. Explain how an insurance company might use the above calculations to compute life insurance premiums. 2. Probability Density Functions: Uniform, Exponential, Normal, and Beta 3. Mean, Median, Variance and Standard Deviation Calculus and Probability Main Page "Real World" Page We would welcome comments and suggestions for improving this resource. Mail us at: Stefan Waner ([email protected]) Steven R. Costenoble ([email protected]) Last Updated: September, 1996

crawl-data/CC-MAIN-2018-51/segments/1544376824115.18/warc/CC-MAIN-20181212181507-20181212203007-00085.warc.gz

First National Government of New Zealand |This article needs additional citations for verification. (May 2010)| The First National Government of New Zealand was the government of New Zealand from 1949 to 1957. It was a conservative government best remembered for its role in the 1951 waterfront dispute. It also began the repositioning of New Zealand in the cold war environment. Although New Zealand continued to assist Britain in situations such as the Malayan Emergency, it now became connected to Australia and the United States through the ANZUS agreement. Domestically, the First National Government presided over a steady rise in the average standard of living, and by 1957 New Zealand was, in the words of the historian Keith Sinclair, “a materialist's paradise.” In 1957, the National Party published a book entitled “A Record of Achievement: The Work of the National Government, 1949-1957,” detailing its accomplishments in office. Under National’s leadership, according to the publication, people now had more money, pensions, cattle, sheep, university scholarships, overseas trips, radios, washing machines, vacuum cleaners, electric toasters, houses, motor vehicles, and telephones. As summed up by Sidney Holland in a foreword, ‘New Zealand is a happier, healthier and more prosperous nation’. - Abolished the Legislative Council (Upper House), thus making New Zealand's parliament unicameral; see Suicide squad. - Established the position of Deputy Prime Minister. - Took the side of employers in the 1951 waterfront dispute - Post-war rationing and price controls on property abolished. - Producer-controlled export boards created. - Set up PAYE income tax. - Formed a partnership with Fletcher Construction to build a pulp and paper mill at Kawerau. Foreign affairs and military This period marked a shift in New Zealand's foreign policy. Before World War II New Zealand lacked an independent foreign policy, instead opting to simply follow and support Britain. New Zealand's participation in World War II was part of this - Prime Minister Michael Joseph Savage had declared that 'where Britain goes we go', and New Zealand troops had fought almost exclusively in Europe rather than in the Pacific, where Japanese forces threatened New Zealand. At the start of the war it had been assumed that the Royal Navy would protect New Zealand, but the Fall of Singapore showed this to be a false assumption. New Zealand turned to the United States for protection. The beginning of the Cold War, and communist successes in China made many New Zealanders feel in need of this protection. New Zealand therefore entered the ANZUS pact with Australia and the United States, each pledging to defend the others if they were attacked. Fear of the communist threat from Asia also motivated the introduction of compulsory military training and New Zealand's participation in the Korean War and the Malayan Emergency. However there was still considerable support for Britain, which led to New Zealand giving Britain moral support (but no practical help) during the Suez Crisis. The government maintained the welfare state created by the previous, Labour, government due to its popularity with voters. However some modifications were made, such as allowing state housing tenants to purchase their homes and enabling families to capitalise their family benefits in order to buy a house. In 1950, the suspensory loan was introduced, a subsidy towards the construction of a home which was repayable if the house was sold within seven years. A year later, universal superannuation was doubled, and a noncontributory social assistance scheme for the underprivileged was introduced. In 1954, widows' benefit was extended to deserted wives after divorce in some cases. The National Party was formed in 1936, after the Labour Party took office for the first time, displacing the Liberal-Reform coalition. The Liberal and Reform parties (along with the Country Party) officially merged into the National Party, initially basing themselves on opposition to Labour and its welfare state policies. However the popularity of these policies soon became evident, and National began to moderate its opposition, promising that it would not abolish the welfare system Labour had enacted. By 1949, Labour had been in power for 14 years. Labour's interventionist ethos combined with the economic restrictions caused by World War II meant that the economy was highly regulated and consumer choice limited. National campaigned on the promise that it would keep the overall structure of Labour's welfare state while moderating it in order to reduce the power of trade unions, increase consumer choice and generally abolish unnecessary regulation. On a relatively small swing, National gained eight seats and became the government for the first time. The 1951 election This was a snap election called after the 1951 waterfront dispute. The dispute was an industrial conflict between the dockworkers' (watersiders') union and the Waterfront Industry Commission, representing employers. Union members had refused to do overtime and had been locked out of the wharves. The dispute lasted from February to July - 151 days. During this time the army was brought in to work the wharves. Prime Minister Sidney Holland argued that militant unions should not be allowed to disrupt the shipping of New Zealand's vital agricultural exports, and the government enacted a range of drastic measures aimed at crushing the union. It was illegal to publish anything in support of the union, or to provide food or other support for the watersiders. The Labour opposition equivocated on the issue, with leader Walter Nash annoying both sides by saying he was 'neither for nor against' the watersiders. The 1951 election was called in order to provide the government with a mandate for its actions during the strike. This was a successful move, as the government was returned with an increased majority. The 1954 election Although National's share of the vote declined significantly from its 1951 levels, it was able to retain its hold on government. This was primarily because both it and Labour had lost votes to the new Social Credit party. With the economy booming, National campaigned on a platform of 'steady as she goes' - simply maintaining the status quo. The major issue in this election was the introduction of PAYE (pay as you earn) income tax. Although both parties were committed to the introduction of the system, they differed in terms of how the changeover from the previous system would be managed. National proposed a complicated rebate system while Labour simply promised a £100 rebate for all taxpayers on the commencement of the new system. Although denounced by National as a bribe, Labour's proposal was the more popular. In addition, National was suffering from leadership problems. Holland appeared old and frail, even compared to Labour leader Walter Nash, who was actually eleven years older. Holland was persuaded to step down from the leadership in favour of Keith Holyoake, but the transition occurred too soon before the election, and Holyoake had little time to establish his leadership. Labour was able to win 4% more of the vote than National, and a slender two seat majority. |Election||Parliament||Seats||Total votes||Percentage||Gain (loss)||Seats won||Change||Majority| Sidney Holland was the Prime Minister for most of the government's term, from 13 December 1949. On 20 September 1957 - less than three months before the election - he stepped down in favour of Keith Holyoake, who was only Prime Minister to 12 December 1957. |Deputy Prime Minister||Keith Holyoake||1949–1957| |Minister of Defence||Tom Macdonald||1949–1957| |Minister of Education||Ronald Algie||1949–1957| |Minister of Finance||Sidney Holland||1949–1954| |Minister of Foreign Affairs||Frederick Doidge||1949–1951| |Minister of Health||Jack Watts||1949–1951| |Minister of Justice||David Thomson||1975–1978| |Minister of Māori Affairs||Ernest Corbett||1949–1957| |Minister of Railways||William Goosman||1949–1954| |Minister of Works||Bill Sullivan||1949–1957| - A History Of New Zealand by Keith Sinclair - Poverty and Progress in New Zealand: A Re-assessment by William Ball Sutch - The Politics of Social Security: The 1938 Act and Some Later Developments by Elizabeth Hanson

<urn:uuid:f241c555-8fd2-4a06-a750-7598edfc7c85>

Our English curriculum ensures that all pupils: - read easily, fluently and with good understanding - develop the habit of reading widely and often, for both pleasure and information - acquire a wide vocabulary, an understanding of grammar and knowledge of linguistic conventions for reading, writing and spoken language - appreciate our rich and varied literary heritage - write clearly, accurately and coherently, adapting their language and style in and for a range of contexts, purposes and audiences - use discussion in order to learn; they should be able to elaborate and explain clearly their understanding and ideas - are competent in the arts of speaking and listening, making formal presentations, demonstrating to others and participating in debate Whole School English The ability to read, write and to communicate well enables all other areas of the curriculum to become accessible. We aim to put English at the heart of our curriculum by using English skills to research other subjects, and by using other subjects as the purpose for English work. English lessons are comprised of a balance of reading and writing, as well as speaking and listening and are taught every day. In Reception and Key Stage . A wide variety of carefully differentiated texts are used to support the children in their development in reading. The children are encouraged to listen attentively, to speak clearly and to express themselves confidently through conversation, class discussion and drama. Our children love sharing their ideas, thoughts and feelings with their learning partners. Pupils are given opportunities to read from a wide range of material for enjoyment and to locate information. All children are encouraged to read daily in and out of school. Pupils are taught to write for a variety of purposes and audiences, producing both fiction and non-fiction genres. They are given opportunities to explain, recount events and to express ideas, thoughts and feelings. We support children in their development of a legible, fluent style of handwriting.

<urn:uuid:a9046546-5ace-4134-baaf-ba0af6a423d2>

…to educate students to be self-directed learners, collaborative workers, complex thinkers, quality producers and community contributors using music study and performance as a framework to experience, understand and demonstrate these ideals. …that the study and performance of music develops and reinforces the cognitive skills and learning strategies that are universal in all fields of academic study. …that the performance music experience engenders the social values and behaviors necessary to meet the district beliefs of dignity, individuality, responsible citizenship, and a collaborative relationship with our community. …that through participation in music performance students are exposed to a unique and powerful method of developing an aesthetic sense and opinion that has positive lasting implications beyond music participation. Music is known as the universal language capable of bridging both time and culture through a unique and wonderful experience for both the creators and the consumers of the art form. Music is a discipline that combines literacy skills with physical development and creative thought with convention. It is an endeavor steeped in tradition yet its survival and testament is born of creativity and multifarious practice. Music is an integral component of all cultures core fabric regardless of time and development and offers a perspective on most societies that is as illuminating as it is entertaining. District 203 will provide a comprehensive and cohesive program of study that develops the necessary skills for music performance. Moreover, the student experience will be saturated with opportunities designed to develop their own individual aesthetic and creative abilities. Our goals are: - To develop students who can successfully perform music in a variety of styles for a variety of audiences. - For students to be able to articulate the emotional and cognitive impact of music. - That all students develop the skills necessary to judge the quality and social and cultural impact of music. - All participating students will acquire a comprehensive foundation of music literacy. Each music classroom experience will provide individualized skill development to equip each student to become a self-directed learner. All music classes will offer an environment rich in investigation, problem solving, experimental inquiry, and creative risk taking. The district music curriculum provides a sequential and evenly paced dissemination of material that will provide for individual student success. The District’s collective attitude about music performance is that teachers must connect with each student at both an intellectual and emotional level. While learning music may be difficult and progress requires effort, enjoyment is an integral thread to the daily activities in all performance music classrooms. Our intent is to instill in all participants a love and respect for music that will stay with every child their entire life.

<urn:uuid:ea0f8e80-d495-42e2-9104-d3af16e584d6>

Researchers at the University of California, Berkeley, announced yesterday that they were able to construct a prism that bent light “the wrong way” and so would make an object appear to vanish [Times UK]. Details of the two different experiments will be published separately later this week in Nature and Science. To bend the light, the scientists used “metamaterials,” mixtures of metal and circuit board materials such as ceramic, Teflon or fiber composite [AP]. One group built what they called a metal “fishnet” of alternating silver and magnesium fluoride; the other used tiny silver nanowires. Both created negative refraction: Light is neither absorbed nor reflected by the objects, passing “like water flowing around a rock,” according to the researchers. As a result, only the light from behind the objects can be seen [BBC]. Two years ago, a team at Duke University created negative refraction with microwaves, but not visible light. Other scientists have used visible light, but could only create negative refraction with two-dimensional objects the thickness of a single atom. The UC Berkeley scientists did two things to get around those problems. First, in order to refract visible light, the nanowires have to be built closer together than its wavelength. Because visible light has a shorter wavelength than microwaves, the UC Berkeley team had to build their materials on a smaller scale than previous researchers had. Second, they layered the materials to a greater thickness in order to achieve negative refraction around a three-dimensional object. But the scientists say scaling up to anything larger will take much more research: For the moment, the vanishing act takes place on a nanoscale, measured in billionths of a metre [AFP]. And the new approaches still only work at limited wavelengths of light, not all of them. So while this research is a step forward toward dreams of invisibility, that’s still far off, according to researcher Jason Valentine. “We are not actually cloaking anything,” Valentine said in a telephone interview. “I don’t think we have to worry about invisible people walking around any time soon. To be honest, we are just at the beginning of doing anything like that” [Reuters].

<urn:uuid:b8699d5f-45ca-453e-8a14-bae4c450f3a9>

# VECTOR MULTIPLICATION BY A SCALAR/UNIT VECTOR FURTHER MATHEMATICS SS 1 SECOND TERM WEEK 5 DATE(S)………………………………………. SUBJECT: FURTHER MATHEMATICS CLASS: SS1 TOPIC: VECTOR 2 CONTENT: 1. Scalar multiplication of vectors 2. Unit vector 3. Direction cosines 4. Scalar (dot) product: Application of scalar (dot) product. 5. Projections of vectors 6. Application of scalar product Sub-Topic: VECTOR MULTIPLICATION BY A SCALAR/UNIT VECTOR Let say and m are scalars then the following laws are true for any vectors a and b: a is also a vector (a+ b) = a + b (distributive) ( + m)a = a + ma. If k is a scalar, then ka is a vector which is parallel to a but k times the magnitude of a. If k>0 then ka is in the same direction of a. However, if k<0, then ka is in a direction opposite to a THE UNIT VECTOR The unit vector is an important concept in the study of vectors. The definition of unit vector was given earlier as a vector which has an absolute value of unity We now amplify on this concept. If OP = ai + bj, then we represent a unit vector in the direction of p by . since a unit vector has a magnitude equal to unity i.e and ll = 1 ll = . Thus a unit vector in the direction of is given by Example 1: Given that , , . Find a unit vector in the direction of and write down a vector parallel to Solution: = Unit vector in the direction of = Example 2: Given that , , , where and are scalars, find an. If + . Solution: + 2( CLASS ACTIVITY 1. Given that the vectors and are parallel, find the constant 2. Given that and , find the unit vector in the direction of Sub-Topic: PROJECTION OR RESOLUTION OF VECTORS b O N A B If the position vector of the point A relative to a reference point o is a and that of the point B relative to O is b, we call the length ON the projection of the vector b on the vector a. If a denotes the unit vector in the direction of the vector a then a =/ậ/ and /ậ/ = 1 Now /on/ = /OB/ cos Ө = /b/ cos Ө =/ȃ/ /b/ cos Ө Then the projection of the vector b on a is ȃ •b where ȃ is the unit vector in the direction of the vector a. Also the projection of the vector a on b is a.Where is the unit vector in the direction of the vector b. As the resolved part of a in the direction of b the projection of the vector a on b can also be viewed. Examples: Find the projection of the vector p on the vector qif: SOLUTION: 1. Let the projection of q on p be u, then U1= / P/ = = p.q = (2)(-1) + (3)(4) = -3 + 12 = 9 Therefore, U1 = = 1. Let the projection q on p be, U2, then U2 = /p/ = = p.q = (4)(-1) + (-5)(1) = -9 Therefore, U2 = X = 1. Let the projection q on p be U3, then U3 = /P/ = = p.q = (6)(2) + (2)(3) = 12 + 6 = 18 Therefore, U3 = X = 1. Let the projection of q and p be U4, then U4 = /p/ = = p.q = (3)(1) + (2)(3) = 3+6 = 9 Therefore, U4 = X = CLASS ACTIVITY For each pair of the following, find the projection of the second vector on the first vector: 1. a = 3 – 4 and b = 5 + 8 2. m = 7+ 3 and n = 2–5 3. x = 4- 5 andy = 3 – 7 SUB TOPIC:- SCALAR (DOT) PRODUCT/DIRECTION COSINES Scalar Product: The scalar product of two vectors a and b is written a.b and is defined as the product of the lengths of the two vectors and the cosine of the angle between them. Thus if Ө is the angle between the vectors a and b then a.b =/a/ Let and be two unit vectors which are perpendicular to each other. Let a= ax + ay b=bx + by Then a.b = (ax + ay) . (bx + by) = axbx . + axby. + aybx. + ayby . As and are unit vectors which are mutually perpendicular to each other we have 1. i. i = 1X1 cos00 = 1 2. j.j = 1 X 1 cos00 = 1 3. i.j =1 x 1 cos 900 =0 4. j.i = 1 x 1 cos 900 = 0 So a.b =axbx + ayby The scalar product can be sometimes be called the dot product. The angle between two vectors Ө can be defined from the definition of scalar product. Thus from a.b = /a/ /b/ cosӨ , we have Cos Ө = If a and b are parallel then Ө = 0 and a.b =/a/ /b/ If a and are perpendicular then Ө = and a.b =0 Example: 1. Find the dot product of the following pairs of vectors: 2. a = + 4 and b = 5 -3 3. p = 6 – and q = 2 – 8 SOLUTION: 1. a.b = ( + 4j) . (5 – 3 =(1)(5) + (4)(-3) = 7 1. p.q = (6i – j ) . (2 – 8 ) = (6)(2) + (-1)(-8) = 20 1. Find the cosine of the angle between the following pairs of vectors; 2. m = 4 + 3 and n= 2 + 5 3. S = 7 – 4 and t = 3 – 2 SOLUTION: 1. Let the angle between the vectors m and n be 𝝰1 then /m/ = /m/ = =5 /n/ = = mn = (4)(2) + (3)(5) = 23 Cos𝝰1= = X = = (b). Let the angle between the vectors s and t be then /s/ = = /t/ = = s.t = (7)(3) + (-4)(-2) = 29 Cos𝝰2 = = = X = = PROPERTIES OF A SCALAR (DOT) PRODUCT 1. Commutative property Let a = ax + ay , b = bx + by Then a.b =b.a 1. Distribution property 2. if //. In particular 2//2 3. if then 4. Multiplication by a scalar (𝛌 5. If ⍬ is the angle between and then cos ⍬= Condition for parallelism If the vector x+ ay is parallel to the vector b =bx + by , i.e =𝛌is a scalar) then = = 𝛌 Condition for perpendicularity If b, i.e then xx + yy = 0 Example: 1. Given that /p / =3 ,/q/ =4 and =-6, find the angle between p and q (WAEC) SOLUTION: Let ⍬ ⍬ = cos-1 -0.5 = 1200. 1. If = 3- 4 and b = 6 – 8 find the scalar product of and SOLUTION: DIRECTION COSINES The direction cosines of vector are: Where α, β and λ are angles which OR makes with OX, OY and OZ axes respectively, where O is the origin and X,Y and Z are mutually perpendicular directions in a 3 – dimensional plane. CLASS ACTIVITY 1. Evaluate: 2. Given that: 3. Find the direction cosines of vectors: (i) (ii) SUB – TOPIC: APPLICATION OF SCALAR PRODUCT Example 1: The vertices A,B and C of a triangle have position vectors and respectively , relative to the origin .prove that . Solution: a B A C b c From the triangle above, taking A is the origin and applying dot product we get By triangle law, a = bc, and a.a = (bc).(bc) Substituting for , we get Example 2: A particle moves from a point with position vector to a point with position vector. A constant force, among other forces acting on the particle is responsible for the movement. Find the work done by the force. Solution: work done is defined as the product of force and displacement, and it is a scalar quantity. work done by , where refers to the displacement of the particle. Work done by = 10 + 24 = 34joules PRACTICE EXERCISE 1. Find the unit vector direction of vectors: (a) (b) 2. Find modulus of each of the following vectors :(i) (ii) (iii) 3. Find the dot product of the following pairs of vectors:(i) 4. If U,V are any two vectors, prove that . Give a geometrical interpretation of this result when: (i) (ii) 5. For what values of λ are the vectors and perpendicular ASSIGNMENT 1. Find the values of µ and λ so that the forces may be in equilibrium. 2. For the following pair of vectors, find the projection of the second vector on the first vector: a = 3 – 4 and b = 5 + 8 3. What is the unit vector which is parallel to the vector ? 4. If m = 3 – 4 and n = 5 + 8 what is the cosine of the angle between m and n ? 5. Given that a= -3 + 4 and b = 2 -, evaluate KEYWORDS • PROJECTION • SCALAR/DOT PRODUCT • MODULUS/MAGNITUDE • UNIT VECTOR (Visited 5 times, 1 visits today) error: Alert: Content is protected !!

crawl-data/CC-MAIN-2021-39/segments/1631780057403.84/warc/CC-MAIN-20210922223752-20210923013752-00679.warc.gz

Stabilization of Curve Directions Let me start from what I’d like to make: I.e., we got two curves, between which I’d like to, for example, stretch Blend or do the previously described center curve procedure by spreading points in Ratio Mode, connect them with lines, and through center points of those lines lead Spline (our approximate center curve). Those two curves have opposite vectors (I’ll purposefully start from such example), meaning, when you indicate them using the tree, Blend won’t pass through without correcting vector of one of the curves… That will give us: So the condition for the procedure is, to direct two curves so that they have the same direction. When marking out center curve, we’ll get false curve, because the straight lines will intersect each other. And something like that will always happen during any change in Parent orientation. How do we avoid that? We need to carry out a procedure, that will somehow connect vectors of those two curves, and keep them together, pointing in the same direction… Here’s an example procedure: – create a Spine curve between our two curves, it will define common direction for both of our curves – next, create normal plane for the Spine: Now let’s stabilize the first of our curves: – create intersections between the normal plane for Spine and the one from our curves, – using that intersection, create normal line for the plane, let’s say 20 mm long (that line will adopt the orientation of the plane), – create a line outgoing from intersection and contacting our curve – keep the 20 mm length (the line will adopt the orientation of our curve), – then, on both lines create a point in Ratio mode (on the ends) : – then create inverse out of our curve, and parameterize it with the following formula: the distance between two points > 20 mm – do the same procedure with the other curve. Now we got two, so to speak, “intelligent” inverses, which react to the change in Parent orientation, and are interconnected through the Spine curve. That will result in two curves with vectors pointing in the same direction… Now we can stretch Blend over, and it will never reverse. Here is a clip showing that procedure: In the end, they’ve developed a template for Blend – for a stable one. And what’s the advantage of that template?? You don’t have to mind the directions of inputs when inserting a template – all you need to do is blindly click them out, and the stabilization procedure will do the rest… A good early approach to templates, at the stage of their creation, i.e., by stabilizing all directions of input elements. An example of such template is already included in the entry about pipe axis – follow that procedure, because there’s a direction that’s been stabilized in a similar way… Soon, I’ll describe how to make a stable parallel curve, and finally, a spatial middle curve (having half of the job already done :-).

crawl-data/CC-MAIN-2020-24/segments/1590347410535.45/warc/CC-MAIN-20200530231809-20200531021809-00571.warc.gz

# Analyzing Teen Texting: Confidence Interval & Plausibility Chapter 8, Problem 36 (choose chapter or problem) Get Unlimited Answers! Check out our subscriptions QUESTION: A Pew Internet and American Life Project survey found that 392 of 799 randomly selected teens reported texting with their friends every day. (a) Calculate and interpret a 95% confidence interval for the population proportion $$p$$ that would report texting with their friends every day. (b) Is it plausible that the true proportion of American teens who text with their friends every day is 0.45? Use your result from part (a) to support your answer. ##### Analyzing Teen Texting: Confidence Interval & Plausibility Want To Learn More? To watch the entire video and ALL of the videos in the series: Discover how to calculate the 95% confidence interval for a population proportion using sample data. Through a practical example about texting habits among American teens, learn the implications of this statistical tool and how to assess plausibility. QUESTION: A Pew Internet and American Life Project survey found that 392 of 799 randomly selected teens reported texting with their friends every day. (a) Calculate and interpret a 95% confidence interval for the population proportion $$p$$ that would report texting with their friends every day. (b) Is it plausible that the true proportion of American teens who text with their friends every day is 0.45? Use your result from part (a) to support your answer. Step 1 of 3 (a) It is required to estimate the population proportion p of teens who reported texting with their friends daily with 95% confidence. Let $$p$$ be the sample proportion of teens who reported texting with their friends every day, as written below: $$\begin{array}{l} p=\frac{392}{799} \\ p=0.49 \end{array}$$ PLAN: To use a one-sample z-interval for $$p$$, the conditions are checked below: 1. Random: The data of teens have been selected randomly. 10% condition: The sample size n has a value of 799 is less than 10% of the population of all teens. 2. Large counts: The expression can be calculated as, \begin{aligned} n p & =799 \times 0.49 \\ & \approx 392 \\ & \geq 10 \\ n(1-p) & =799 \times(1-0.49) \\ & \approx 407 \\ & \geq 10 \end{aligned}

crawl-data/CC-MAIN-2023-50/segments/1700679102612.80/warc/CC-MAIN-20231210155147-20231210185147-00172.warc.gz

In the visual observing situation, the observer is the human eye that receives the light reflected from or transmitted through an object, and the brain which perceives the vision. Since different humans perceive color and appearance in different ways, subjectively, attempts have been made to “standardize” the human observer as a numerical representation of what the “average person” sees. This standard observer could then be used in lieu of a human observer when assessments are made instrumentally. Wright and Guild performed experiments using human volunteers to assess their color vision and develop an average, or standard, observer. In 1931 they published the 2° CIE Standard Observer function based on their research. The function is called 2° because their experiments involved having the subjects judge colors while looking through a hole that allowed them a 2° field of view. In 1931, it was believed that all the color-sensing cones of the eye were located within a 2° arc of the fovea. Thus the 2° field of view was chosen and used in establishing the standard observer. By the 1960s, it was realized that cones were present in a larger area of the eye than previously believed, and so in 1964, the 10° Standard Observer was developed. The 10° Standard Observer is currently believed to best represent the average spectral response of human observers, although the 2° Standard Observer still has its place for measurement of objects that will be viewed at a distance, such as road signs. The relative sizes of the two fields of view are shown below. The standard observers, in the form of mathematical functions of the human response to each wavelength of light, are used in color calculations. The observers can be graphed as shown below. The CIE Tristimulus XYZ color scale, for instance, is calculated as follows: X = ∫(R or T) * illuminant factor * x factor of standard observer Y = ∫(R or T) * illuminant factor * y factor of standard observer Z = ∫(R or T) * illuminant factor * z factor of standard observer R = % reflectance T = % transmittance Sums are across the spectral range for which the instrument reads. Note that X, Y, and Z include factors for the mathematical standard observer in their formulas. All other tristimulus color scales (such as Hunter L, a, b and CIEL*a*b*) may then be calculated from XYZ. (See attached pdf file for the complete article with illustrations)

<urn:uuid:fd72dbed-a42f-4dcf-abb0-c65fb22fda5c>

# How do you solve 6a+5a=-11? Aug 3, 2016 $a = - 1$ #### Explanation: $6 a + 5 a = - 11$ or $11 a = - 11$ or $a = - \frac{11}{11}$ or $a = - 1$

crawl-data/CC-MAIN-2019-51/segments/1575540527205.81/warc/CC-MAIN-20191210095118-20191210123118-00237.warc.gz

noun, plural: chromomeres In genetics, chromomere is one of those beadlike granules arranged in a linear series on the chromosomes of eukaryotes. Chromomeres form from the local coiling of a continuous DNA thread. They become more distinct during prophase of both mitosis and meiosis. In meiosis, they are evident as early as the leptotene phase of prophase I especially because the chromosomes are starting to get condensed at this stage. In the next stage, i.e. zygotene, wherein the homologous chromosomes pair up in pairs, the chromomeres aid the homologous chromosomes to align with each other and form homologous rough pairing. Chromomeres contain genes and the arrangement of chromomere structure may be applied in controlling gene expression. Maps of chromomere structure may be made to be used for genetic as well as for evolutionary studies. They may prove useful in locating genes on a chromosome and in analyzing chromosomal aberrations. In anatomy, chromomere pertains to the central part of a blood platelet. It is also called granulomere because the core of the platelet in stained smears is filled with small purple granules. The granules may be so compact in the central region of the platelet that it may be wrongly perceived as a nucleus. These granules however are evenly distributed in circulating platelets and become compacted from smear preparations. Furthermore, platelets are anucleate bodies. Electron micrographs of the platelet's granulomere reveals the presence of lysosomes, mitochondria, granules comprised of serotonin, ADP, ATP, calcium, alpha particles (with platelet-specific proteins, fibrinogen, and other clotting factors), actin, myosin, ATPase, and agents for increasing vascular permeability.1 Word origin: chromo- (denoting color) + Greek meros (a part) - idiomere (genetics) - granulomere (anatomy) 1 Krause, W. & Krause, W. (2005). Krause's essential human histology for medical students. Boca Raton: Universal Publishers. p.70.

<urn:uuid:d62e6498-531e-4b1c-92f1-9b189a29f421>

# Section 6.6—Limiting Reactants What happens if you don’t add reactants in a molar ratio? ## Presentation on theme: "Section 6.6—Limiting Reactants What happens if you don’t add reactants in a molar ratio?"— Presentation transcript: Section 6.6—Limiting Reactants What happens if you don’t add reactants in a molar ratio? Planning a Meal You go to the grocery store and you buy 1 package of Brats (5 Brats), 1 package of cheese (16 slices) and 1 package of hot dog buns (8 buns). If you use all of these…You can make this many… 5 Brats 5 meals 16 slices of cheese 8 hot dog buns 16 meals 8 meals So you have the possibility of making 5, 16 or 8 meals…which is it? You’ll never get the chance to make 8 or 16 meals…you’ll run out of Brats after 5. Once you run out of one component, you have to stop making meals. What’s a limiting reactant? Limiting Reactant – The reactant that runs out and causes the reaction to stop. In the previous example, the Brats were the limiting reactant—once they were gone, you had to stop! Once even one of the reactants runs out, the reaction stops…it can’t make any more product. Excess reactant The reactant that is not completely used up in a reaction When solving: Both reactants will be given info. Use stoichiometry to find the (mass, moles, vol. etc) of EACH reactant Whichever number is smaller will be your limiting reactant Limiting Reactant Example Example: How many moles of H 2 O is produced when 2.3 moles O 2 and 2.3 moles H 2 react? 2 H 2 + O 2  2 H 2 O From balanced equation: 2 mole H 2  2 mole H 2 O 1 mole O 2  2 mole H 2 O Limiting Reactant Example 2.3 mole O 2 mole O 2 mole H 2 O = ________ mole H 2 O 1 2 4.6 2.3 mole H 2 mole H 2 mole H 2 O = ________ mole H 2 O 2 2 2.3 Example: How many moles of H 2 O is produced when 2.3 moles O 2 and 2.3 moles H 2 react? 2 H 2 + O 2  2 H 2 O From balanced equation: 2 mole H 2  2 mole H 2 O 1 mole O 2  2 mole H 2 O Limiting Reactant Example 2.3 mole O 2 mole O 2 mole H 2 O = ________ mole H 2 O 1 2 4.6 2.3 mole H 2 mole H 2 mole H 2 O = ________ mole H 2 O 2 2 2.3 Example: How many moles of H 2 O is produced when 2.3 moles O 2 and 2.3 moles H 2 react? 2 H 2 + O 2  2 H 2 O Limiting reactant! Let’s Practice Example: If you react 10.5 g of NaOH and 7.5 g of BaCl 2, how many grams NaCl is produced? 2 NaOH + BaCl 2  Ba(OH) 2 + 2 NaCl From balanced equation: 2 mole NaOH  2 mole NaCl 1 mole BaCl 2  2 mole NaCl Let’s Practice 10.50 g NaOH g NaOH mole NaOH = _______ g NaCl 40.00 1 15.3 Molecular masses: 40.00 g NaOH = 1 mole NaOH 171.35 g Ba(OH) 2 = 1 mole Ba(OH) 2 58.44 g NaCl = 1 mole NaCl mole NaOH mole NaCl 2 2 g NaCl 1 58.44 Example: If you react 10.5g of NaOH and 7.5g of BaCl 2, how many grams NaCl is produced? 2 NaOH + BaCl 2  Ba(OH) 2 + 2 NaCl 7.50 g BaCl 2 g BaCl 2 mole BaCl 2 = _______ g NaCl 171.35 1 5.12 mole BaCl 2 mole NaCl 1 2 g NaCl 1 58.44 Lets Practice 2Na + Cl 2  2NaCl Suppose that 6.70 mol of Na reacts with 3.20 mol of Cl 2.  1) What is the limiting reactant?  How many moles of NaCl are produced? Lets Practice 2Cu (s) + S (s)  Cu 2 S (s)  What is the limiting regent when 80.0g Cu reacts with 25.0g S?  How grams of Cu 2 S will be produced?

crawl-data/CC-MAIN-2017-43/segments/1508187824357.3/warc/CC-MAIN-20171020211313-20171020231313-00227.warc.gz

# Approximating Euler’s number correctly ## Introduction Suppose we want to calculate $$e$$ (Euler’s number, Napier’s constant, 2.718281828...) accurate to 1000 decimal places. How can we do this from scratch with only big integer support, without the help of a computer algebra system? The infinite series definition taught in introductory calculus is a good place to start at. But how many terms do we need to add up before truncating the series? How do we know the error bounds so that we can say for sure the result is correctly rounded? Do we need extra precision for intermediate calculations? Here is a sketch of some inadequate code: double sum = 0.0; double factorial = 1.0; for (int i = 0; i < 99; i++) { // When to terminate series? sum += 1 / factorial; // How much error accumulated? factorial *= i + 1; // Rounding error? } On this page we will go through the mathematics and algorithms to calculate $$e$$ correctly from first principles, practical up to about 100 000 digits. ## Basic definitions Our basis will be the textbook definition of the number $$e$$: $$e \: = \: \displaystyle\sum_{k=0}^∞ \frac{1}{k!} \: = \: \frac{1}{0!} + \frac{1}{1!} + \frac{1}{2!} + \frac{1}{3!} + \cdots .$$ Define the sequences of partial sums and remainders, for all $$n ∈ \mathbb{N}$$: $$S_n \: = \: \displaystyle\sum_{k=0}^n \frac{1}{k!} \: = \: \frac{1}{0!} + \frac{1}{1!} + \frac{1}{2!} + \cdots + \frac{1}{n!} . \\ R_n \: = \: e - S_n \: = \: \frac{1}{(n + 1)!} + \frac{1}{(n + 2)!} + \frac{1}{(n + 3)!} + \cdots .$$ ## Main theorem Because all the terms in the sum are strictly positive, it’s clear that every $$R_n > 0$$. Now let’s derive an upper bound on an arbitrary $$R_n$$, assuming that $$n ≥ 1$$: \begin{align} R_n \: &< \: \frac{1}{(n + 1)!} + \frac{1}{(n + 1)! \: (n + 1)} + \frac{1}{(n + 1)! \: (n + 1)^2} + \cdots \\ &= \: \frac{1}{(n + 1)!} \left[ 1 + \frac{1}{n + 1} + \frac{1}{(n + 1)^2} + \cdots \right] \\ &= \: \frac{1}{(n + 1)!} \displaystyle\sum_{k=0}^∞ \frac{1}{(n + 1)^k} \\ &= \: \frac{1}{(n + 1)!} \frac{n + 1}{n} \: = \: \frac{1}{n! \: n} \\ &\leq \: \frac{1}{n!}. \end{align} Explanations: • The first line is due to the definition of the factorial function, like $$(n + 2)! = (n + 1)! \: (n + 2) > (n + 1)! \: (n + 1)$$, etc. • The second line is by factoring like terms. • The third line rewrites the infinite sum formally. • The fourth line is due to the well-known geometric series. • The fifth line weakens the inequality (because $$0 < \frac{1}{n} ≤ 1$$), which will simplify the computation later on. In summary, we have for every $$n ≥ 1$$: $$0 \: < \: R_n \: < \: \frac{1}{n!}.$$ Add $$S_n$$ to all sides to get: $$S_n \: < \: S_n + R_n = e \: < \: S_n + \frac{1}{n!}.$$ Or negate the inequality and add $$e$$ to get: $$e - \frac{1}{n!} \: < \: S_n \: < \: e.$$ In other words, when truncating the infinite series for $$e$$ after $$n+1$$ terms (i.e. the last term is $$\frac{1}{n!}$$), this partial sum $$S_n$$ is strictly less than $$e$$, but differs from $$e$$ by no more than $$\frac{1}{n!}$$. The main idea in the algorithms described below is that when both $$S_n$$ and $$S_n + \frac{1}{n!}$$ are rounded to the same value, we can be sure this is the correct approximation of $$e$$. Otherwise we continue adding terms to the partial sum and wait for the difference between these two values to shrink. ## Fraction algorithm This algorithm follows fairly straightforwardly from the mathematical argument, with $$m$$ being the number of decimal places we want to calculate: 1. Start with $$n = 0$$. 2. (Top of loop) Calculate the partial sum $$S_n$$ as an exact fraction. 3. Consider the inequality $$S_n < e < S_n + \frac{1}{n!}$$, which says that the true value of $$e$$ lies within an interval of length $$\frac{1}{n!}$$. If the interval length exceeds $$10^{-m}$$ then the lower and upper ends of the interval round to different numbers, so no answer is available. 4. If $$\frac{1}{n!} < 10^{-m}$$, then we check whether $$\text{round}(S_n) = \text{round}(S_n + \frac{1}{n!})$$. If equal, then exit the loop and return $$\text{round}(S_n)$$ as the result. 5. Otherwise increment $$n$$ and loop again. If your programming language doesn’t have a library for fractions / rational numbers, it’s not a problem because the functionality can be implemented in a few dozen lines of code. Unfortunately, the fractions get big quickly because the denominator is $$n!$$. In practice, this algorithm takes about 20 seconds to compute 3 000 decimal places on my computer, which is hardly impressive. Source code: ## Interval algorithm Instead of calculating the partial sum and each term using exact fractions, let’s approximate them by using closed intervals (i.e. $$[\text{low}, \text{high}]$$) to represent where the true value must reside. The key idea of this algorithm is to use fixed-point arithmetic with {$$m$$ plus an extra $$p$$} decimal places of precision, along with interval arithmetic to bound the uncertainty. So for the low end we round calculations down, and for the high end we round calculations up. This procedure is a refinement of the fraction algorithm with added complexity: 1. Start with $$n = 0$$, $$\text{sum} = [0, 0]$$, $$\text{term} = [10^{m+p}, 10^{m+p}]$$. 2. (Top of loop) Let $$\text{sum} = [\text{sum}_L + \text{term}_L, \: \text{sum}_H + \text{term}_H]$$. 3. Because $$\frac{10^{m+p}}{n!} ∈ [\text{term}_L, \text{term}_H]$$, we know that $$\text{sum}_L < e < \text{sum}_H + \text{term}_H$$. 4. If $$\text{term}_H < 10^p$$ (analogous to $$\frac{1}{n!} < 10^{-m}$$), then it may be possible to generate a result. In particular, if $$\text{round}(\text{sum}_L) = \text{round}(\text{sum}_H + \text{term}_H)$$, then we return this as the result. 5. Otherwise increment $$n$$, let $$\text{term} = [\lfloor \text{term}_L / n \rfloor, \: \lceil \text{term}_H / n \rceil ]$$, and loop again. This algorithm is much faster than the fraction-based algorithm, taking about 20 seconds on my computer to get 100 000 decimal places (30× more digits for the same time spent). Actually, step 5 can lead to a number of simplifications: • Truncating division is available but ceiling division usually isn’t, so we can be lazy by setting $$\text{term}_H = \lfloor \text{term}_H / n \rfloor + 1$$ (since pessimistically $$\lceil x \rceil ≤ \lfloor x \rfloor + 1$$). • We can be lazier and more pessimistic by fixing $$\text{term}_H = \text{term}_L + n + 1$$ (because $$n$$ floor operations were performed, and each division by a positive integer does not increase the error). • Finally, we can be laziest by fusing $$\text{term}_H$$ into $$\text{sum}_H$$ by always letting $$\text{sum}_H = \text{sum}_L + \frac{n(n+1)}{2}$$ (due to the arithmetic series). Source code: ## The exponential function We can extend this analysis and approximate the exponential function using the same line of reasoning. For simplicity, assume that $$x > 0$$. Recall the standard definition: $$\exp(x) \: = \: \displaystyle\sum_{k=0}^∞ \frac{x^k}{k!} \: = \: \frac{x^0}{0!} + \frac{x^1}{1!} + \frac{x^2}{2!} + \frac{x^3}{3!} + \cdots .$$ Define the partial sums and the remainders in the same way, for all $$n ∈ \mathbb{N}$$: $$S_n(x) \: = \: \displaystyle\sum_{k=0}^n \frac{x^k}{k!} \: = \: \frac{x^0}{0!} + \frac{x^1}{1!} + \frac{x^2}{2!} + \cdots + \frac{x^n}{n!} . \\ R_n(x) \: = \: \exp(x) - S_n(x) \: = \: \frac{x^{n+1}}{(n + 1)!} + \frac{x^{n+2}}{(n + 2)!} + \frac{x^{n+3}}{(n + 3)!} + \cdots .$$ It should be clear that $$R_n(x) > 0$$ for all $$x > 0$$ and $$n ∈ \mathbb{N}$$. Now let’s derive the main inequality, assuming that we pick an integer $$n$$ such that $$n > x$$: \begin{align} R_n(x) \: &< \: \frac{x^{n+1}}{(n + 1)!} + \frac{x^{n+2}}{(n + 1)! \: (n + 1)} + \frac{x^{n+3}}{(n + 1)! \: (n + 1)^2} + \cdots \\ &= \: \frac{x^{n+1}}{(n + 1)!} \left[ 1 + \frac{x}{n + 1} + \frac{x^2}{(n + 1)^2} + \cdots \right] \\ &= \: \frac{x^{n+1}}{(n + 1)!} \displaystyle\sum_{k=0}^∞ \left( \frac{x}{n + 1} \right)^k \: = \: \frac{x^{n+1}}{(n + 1)!} \frac{n + 1}{n + 1 - x} \\ &= \: \frac{x^{n+1}}{n! \: (n + 1 - x)} < \: \frac{x^{n+1}}{n!}. \\ \end{align} Therefore we have: $$0 < R_n(x) < \displaystyle \frac{x^{n+1}}{n!}. \\ S_n(x) < \exp(x) < S_n(x) + \displaystyle \frac{x^{n+1}}{n!}.$$ Here is the program based on the interval algorithm. Source code: ## Notes • These algorithms work for any rounding mode as long as the rounding is a monotonic function. For example, floor, ceiling, truncation, and round-half-to-even are all acceptable. • I tried to be fairly rigorous and explicit in the mathematical analysis, though I did omit most of the basic algebra. • The other textbook definition of $$e = \displaystyle\lim_{n \to ∞} \left( 1 + \frac{1}{n} \right)^n$$ is not useful for computational purposes. • The algorithms described by me here are nowhere near the state of the art. A casual look at the list of records shows that it’s quite reasonable to compute billions of digits of $$e$$. • There is no guarantee that my algorithms, or anyone else’s algorithms terminate. This is because the value being computed could be very close to a rounding boundary, like in the case of 1.499999999992. However, the chance of this happening for a “nice” number like $$e$$ (which is irrational and possibly normal) is exceedingly small, so this is not a concern in practice. • Computing $$\exp(x)$$ for large $$x$$ requires a lot of extra precision. I am aware of this problem but have no solution to offer. • Handling the $$x < 0$$ case is left as an exercise to the reader. It’ll be somewhat uglier because the sum has positive and negative terms. • My formulas and inequalities for approximating $$e$$ have a lot in common with a proof that $$e$$ is irrational. • Easy-to-follow ideas for speeding up the calculation: Brothers Technology: Improving the Convergence of Newton’s Series Approximation for e

crawl-data/CC-MAIN-2024-38/segments/1725700651548.18/warc/CC-MAIN-20240914025441-20240914055441-00145.warc.gz

$$\require{cancel}$$ # 19.1: Electric Potential Energy- Potential Difference Learning Objectives By the end of this section, you will be able to: • Define electric potential and electric potential energy. • Describe the relationship between potential difference and electrical potential energy. • Explain electron volt and its usage in submicroscopic process. • Determine electric potential energy given potential difference and amount of charge. When a free positive charge $$q$$ is accelerated by an electric field, such as shown in Figure $$\PageIndex{1}$$, it is given kinetic energy. The process is analogous to an object being accelerated by a gravitational field. It is as if the charge is going down an electrical hill where its electric potential energy is converted to kinetic energy. Let us explore the work done on a charge $$q$$ by the electric field in this process, so that we may develop a definition of electric potential energy. The electrostatic or Coulomb force is conservative, which means that the work done on $$q$$ is independent of the path taken. This is exactly analogous to the gravitational force in the absence of dissipative forces such as friction. When a force is conservative, it is possible to define a potential energy associated with the force, and it is usually easier to deal with the potential energy (because it depends only on position) than to calculate the work directly. We use the letters PE to denote electric potential energy, which has units of joules (J). The change in potential energy, $$\Delta \mathrm{PE}$$, is crucial, since the work done by a conservative force is the negative of the change in potential energy; that is, $$W=-\Delta \mathrm{PE}$$. For example, work $$W$$ done to accelerate a positive charge from rest is positive and results from a loss in PE, or a negative $$\Delta \mathrm{PE}$$. There must be a minus sign in front of $$\Delta \mathrm{PE}$$ to make $$W$$ positive. PE can be found at any point by taking one point as a reference and calculating the work needed to move a charge to the other point. POTENTIAL ENERGY $$W=-\Delta \mathrm{PE}$$. For example, work $$W$$ done to accelerate a positive charge from rest is positive and results from a loss in PE, or a negative $$\Delta \mathrm{PE}$$ There must be a minus sign in front of $$\Delta \mathrm{PE}$$ to make $$W$$ positive. PE can be found at any point by taking one point as a reference and calculating the work needed to move a charge to the other point. Gravitational potential energy and electric potential energy are quite analogous. Potential energy accounts for work done by a conservative force and gives added insight regarding energy and energy transformation without the necessity of dealing with the force directly. It is much more common, for example, to use the concept of voltage (related to electric potential energy) than to deal with the Coulomb force directly. Calculating the work directly is generally difficult, since $$W=Fd\cos \theta$$ and the direction and magnitude of $$F$$ can be complex for multiple charges, for odd-shaped objects, and along arbitrary paths. But we do know that, since $$F=qE$$, the work, and hence $$\Delta \mathrm{PE}$$, is proportional to the test charge $$q$$ To have a physical quantity that is independent of test charge, we define electric potential $$V$$ (or simply potential, since electric is understood) to be the potential energy per unit charge: $V=\dfrac{\mathrm{PE}}{q}.$ ELECTRIC POTENTIAL This is the electric potential energy per unit charge. $V=\dfrac{\mathrm{PE}}{q}$ Since PE is proportional to $$q$$, the dependence on $$q$$ cancels. Thus $$V$$ does not depend on $$q$$. The change in potential energy $$\Delta \mathrm{PE}$$ is crucial, and so we are concerned with the difference in potential or potential difference $$\Delta V$$ between two points, where $\Delta V =V_{B}-V_{A}=\dfrac{\Delta \mathrm{PE}}{q}.$ The potential difference between points A and B, $$V_{B}-V_{A}$$, is thus defined to be the change in potential energy of a charge $$q$$ moved from A to B, divided by the charge. Units of potential difference are joules per coulomb, given the name volt (V) after Alessandro Volta. $1\mathrm{V}=1\mathrm{\dfrac{J}{C}}$ POTENTIAL DIFFERENCE The potential difference between points A and B, $$V_{B}-V_{A}$$, is defined to be the change in potential energy of a charge $$q$$ moved from A to B, divided by the charge. Units of potential difference are joules per coulomb, given the name volt (V) after Alessandro Volta. $1\mathrm{V}=1\mathrm{\dfrac{J}{C}}$ The familiar term voltage is the common name for potential difference. Keep in mind that whenever a voltage is quoted, it is understood to be the potential difference between two points. For example, every battery has two terminals, and its voltage is the potential difference between them. More fundamentally, the point you choose to be zero volts is arbitrary. This is analogous to the fact that gravitational potential energy has an arbitrary zero, such as sea level or perhaps a lecture hall floor. In summary, the relationship between potential difference (or voltage) and electrical potential energy is given by $\Delta V=\dfrac{\Delta \mathrm{PE}}{q}\: \mathrm{and}\: \Delta \mathrm{PE}=q\Delta V.$ POTENTIAL DIFFERENCE AND ELECTRICAL POTENTIAL ENERGY The relationship between potential difference (or voltage) and electrical potential energy is given by $\Delta =\dfrac{\Delta \mathrm{PE}}{q}\: \mathrm{and}\: \Delta \mathrm{PE}=q\Delta V.$ The second equation is equivalent to the first. Voltage is not the same as energy. Voltage is the energy per unit charge. Thus a motorcycle battery and a car battery can both have the same voltage (more precisely, the same potential difference between battery terminals), yet one stores much more energy than the other since $$\Delta PE=q\Delta V$$. The car battery can move more charge than the motorcycle battery, although both are 12 V batteries. Example $$\PageIndex{1}$$:Calculating Energy Suppose you have a 12.0 V motorcycle battery that can move 5000 C of charge, and a 12.0 V car battery that can move 60,000 C of charge. How much energy does each deliver? (Assume that the numerical value of each charge is accurate to three significant figures.) Strategy To say we have a 12.0 V battery means that its terminals have a 12.0 V potential difference. When such a battery moves charge, it puts the charge through a potential difference of 12.0 V, and the charge is given a change in potential energy equal to $$\Delta PE=q\Delta V$$. So to find the energy output, we multiply the charge moved by the potential difference. Solution For the motorcycle battery, $$q=5000 \mathrm{C}$$ and $$\Delta =12.0\mathrm{V}$$. The total energy delivered by the motorcycle battery is $\Delta \mathrm{PE}_{cycle}=(5000\mathrm{C})(12.0\mathrm{V})$ $=(5000\mathrm{C})(12.0\mathrm{J/C})$ $=6.00\times 10^{4}\mathrm{J}.$ Similarly, for the car battery, $$q=60,000\mathrm{C}$$ and $\Delta \mathrm{PE}_{car}=(60,000\mathrm{C})(12.0\mathrm{V})$ $=7.20\times 10^{5}\mathrm{J}.$ Discussion While voltage and energy are related, they are not the same thing. The voltages of the batteries are identical, but the energy supplied by each is quite different. Note also that as a battery is discharged, some of its energy is used internally and its terminal voltage drops, such as when headlights dim because of a low car battery. The energy supplied by the battery is still calculated as in this example, but not all of the energy is available for external use. Note that the energies calculated in the previous example are absolute values. The change in potential energy for the battery is negative, since it loses energy. These batteries, like many electrical systems, actually move negative charge—electrons in particular. The batteries repel electrons from their negative terminals (A) through whatever circuitry is involved and attract them to their positive terminals (B) as shown in Figure $$\PageIndex{2}$$. The change in potential is $$\Delta V =V_{B}-V_{A}=+12\mathrm{V}$$ and the charge $$q$$ is negative, so that $$\Delta \mathrm{PE}=q\Delta V$$ is negative, meaning the potential energy of the battery has decreased when $$q$$ has moved from A to B. Example $$\PageIndex{2}$$: How Many Electrons Move through a Headlight Each Second? When a 12.0 V car battery runs a single 30.0 W headlight, how many electrons pass through it each second? Strategy To find the number of electrons, we must first find the charge that moved in 1.00 s. The charge moved is related to voltage and energy through the equation $$\Delta \mathrm{PE}=q\Delta V$$. A 30.0 W lamp uses 30.0 joules per second. Since the battery loses energy, we have $$\Delta \mathrm{PE}=-30.0J$$ and, since the electrons are going from the negative terminal to the positive, we see that $$\Delta V=+12.0V$$. Solution To find the charge $$q$$ moved, we solve the equation $$\Delta \mathrm{PE}=q\Delta V$$: $q=\dfrac{\Delta \mathrm{PE}}{\Delta V}.$ Entering the values for $$\Delta PE$$ and $$\Delta V$$, we get $q=\dfrac{-30.0\mathrm{J}}{+12.0\mathrm{V}}=\dfrac{-30.0\mathrm{J}}{+12.0\mathrm{J/C}}=-2.50\mathrm{C}.$ The number of electrons $$n_{e}$$ is the total charge divided by the charge per electron. That is, $n_{e}=\dfrac{-2.50\mathrm{C}}{-1.60\times 10^{-19}\mathrm{C/e^{-}}}=1.56\times 10^{19} \mathrm{electrons}.$ Discussion This is a very large number. It is no wonder that we do not ordinarily observe individual electrons with so many being present in ordinary systems. In fact, electricity had been in use for many decades before it was determined that the moving charges in many circumstances were negative. Positive charge moving in the opposite direction of negative charge often produces identical effects; this makes it difficult to determine which is moving or whether both are moving. ## The Electron Volt The energy per electron is very small in macroscopic situations like that in the previous example—a tiny fraction of a joule. But on a submicroscopic scale, such energy per particle (electron, proton, or ion) can be of great importance. For example, even a tiny fraction of a joule can be great enough for these particles to destroy organic molecules and harm living tissue. The particle may do its damage by direct collision, or it may create harmful x rays, which can also inflict damage. It is useful to have an energy unit related to submicroscopic effects. Figure $$\PageIndex{3}$$ shows a situation related to the definition of such an energy unit. An electron is accelerated between two charged metal plates as it might be in an old-model television tube or oscilloscope. The electron is given kinetic energy that is later converted to another form—light in the television tube, for example. (Note that downhill for the electron is uphill for a positive charge.) Since energy is related to voltage by $$\Delta PE=q\Delta V$$ we can think of the joule as a coulomb-volt. On the submicroscopic scale, it is more convenient to define an energy unit called the electron volt (eV), which is the energy given to a fundamental charge accelerated through a potential difference of 1 V. In equation form, $1\mathrm{ev}=(1.60\times 10^{-19}\mathrm{C})(1\mathrm{V})=(1.60\times 10^{-19}\mathrm{C})(1\mathrm{J/C})$ $=1.60\times 10^{-19}J.$ ELECTRON VOLT On the submicroscopic scale, it is more convenient to define an energy unit called the electron volt (eV), which is the energy given to a fundamental charge accelerated through a potential difference of 1 V. In equation form, $1 \mathrm{eV}=(1.60\times 10^{-19} \mathrm{C})(1 \mathrm{V})=(1.60\times 10^{-19} \mathrm{C}) (1\mathrm{J/C})$ $=1.60\times 10^{-19} \mathrm{C}$ An electron accelerated through a potential difference of 1 V is given an energy of 1 eV. It follows that an electron accelerated through 50 V is given 50 eV. A potential difference of 100,000 V (100 kV) will give an electron an energy of 100,000 eV (100 keV), and so on. Similarly, an ion with a double positive charge accelerated through 100 V will be given 200 eV of energy. These simple relationships between accelerating voltage and particle charges make the electron volt a simple and convenient energy unit in such circumstances. CONNECTIONS: ENERGY UNITS The electron volt (eV) is the most common energy unit for submicroscopic processes. This will be particularly noticeable in the chapters on modern physics. Energy is so important to so many subjects that there is a tendency to define a special energy unit for each major topic. There are, for example, calories for food energy, kilowatt-hours for electrical energy, and therms for natural gas energy. The electron volt is commonly employed in submicroscopic processes—chemical valence energies and molecular and nuclear binding energies are among the quantities often expressed in electron volts. For example, about 5 eV of energy is required to break up certain organic molecules. If a proton is accelerated from rest through a potential difference of 30 kV, it is given an energy of 30 keV (30,000 eV) and it can break up as many as 6000 of these molecules ( $$30,000 \mathrm{eV}\div 5\mathrm{eV}$$ per molecule $$=6000$$ molecules). Nuclear decay energies are on the order of 1 MeV (1,000,000 eV) per event and can, thus, produce significant biological damage. ## Conservation of Energy The total energy of a system is conserved if there is no net addition (or subtraction) of work or heat transfer. For conservative forces, such as the electrostatic force, conservation of energy states that mechanical energy is a constant. Mechanical energy is the sum of the kinetic energy and potential energy of a system; that is, $$KE + PE=\: \mathrm{constant}$$. A loss of PE of a charged particle becomes an increase in its KE. Here PE is the electric potential energy. Conservation of energy is stated in equation form as $\mathrm{KE}+\mathrm{PE}=\mathrm{constant}$ or $\mathrm{KE}_{i}+\mathrm{PE}_{i}=\mathrm{KE}_{f}+\mathrm{PE}_{f},$ where i and f stand for initial and final conditions. As we have found many times before, considering energy can give us insights and facilitate problem solving. Example $$\PageIndex{3}$$: Electrical Potential Energy Converted to Kinetic Energy Calculate the final speed of a free electron accelerated from rest through a potential difference of 100 V. (Assume that this numerical value is accurate to three significant figures.) Strategy We have a system with only conservative forces. Assuming the electron is accelerated in a vacuum, and neglecting the gravitational force (we will check on this assumption later), all of the electrical potential energy is converted into kinetic energy. We can identify the initial and final forms of energy to be $$\mathrm{KE}_{i}=0,\mathrm{KE}_{f}=\dfrac{1}{2}mv^{2}, \mathrm{PE}_{i}=qV,\: \mathrm{and}\: \mathrm{PE}_{f}=0$$. Solution Conservation of energy states that $\mathrm{KE}_{i}+\mathrm{PE}_{i}=\mathrm{KE}_{f}+\mathrm{PE}_{f}$ Entering the forms identified above, we obtain $qV=\dfrac{mv^{2}}{2}.$ We solve this for $$v$$: $v=\sqrt {\dfrac{2qV}{m}}.$ Entering values for $$q,\: V,\: \mathrm{and}\: m$$ gives $v=\sqrt{\dfrac{2(-1.60\times 10^{-19}\mathrm{C})(-100 \mathrm{J/C})}{9.11\times 10^{-31}\mathrm{kg}}}$ $=5.93\times 10^{6} \mathrm{m/s}.$ Discussion Note that both the charge and the initial voltage are negative, as in Figure. From the discussions in Electric Charge and Electric Field, we know that electrostatic forces on small particles are generally very large compared with the gravitational force. The large final speed confirms that the gravitational force is indeed negligible here. The large speed also indicates how easy it is to accelerate electrons with small voltages because of their very small mass. Voltages much higher than the 100 V in this problem are typically used in electron guns. Those higher voltages produce electron speeds so great that relativistic effects must be taken into account. That is why a low voltage is considered (accurately) in this example. ## Summary • Electric potential is potential energy per unit charge. • The potential difference between points A and B, $$V_{\mathrm{B}}-V_{\mathrm{A}}$$, defined to be the change in potential energy of a charge $$q$$ moved from A to B, is equal to the change in potential energy divided by the charge, Potential difference is commonly called voltage, represented by the symbol $$\Delta V$$. $$\Delta V= \dfrac{\Delta \mathrm{PE}}{q}\: \mathrm{and}\: \Delta \mathrm{PE}=q\Delta V.$$ • An electron volt is the energy given to a fundamental charge accelerated through a potential difference of 1 V. In equation form, $$1\mathrm{eV}=(1.60\times 10^{-19}\mathrm{C})(1 \mathrm{V})=(1.60\times 10^{-19}\mathrm{C})(1 \mathrm{J/C})$$ $$=1.60\times 10^{-19}\mathrm{J}.$$ • Mechanical energy is the sum of the kinetic energy and potential energy of a system, that is, $$\mathrm{KE}+\mathrm{PE}$$ This sum is a constant. ## Glossary electric potential potential energy per unit charge potential difference (or voltage) change in potential energy of a charge moved from one point to another, divided by the charge; units of potential difference are joules per coulomb, known as volt electron volt the energy given to a fundamental charge accelerated through a potential difference of one volt mechanical energy sum of the kinetic energy and potential energy of a system; this sum is a constant

crawl-data/CC-MAIN-2021-49/segments/1637964363445.41/warc/CC-MAIN-20211208053135-20211208083135-00292.warc.gz

The Prince Albert Mountains rise up from Antarctica’s Scott Coast on the western shores of the Ross Sea. Like the Rocky Mountains, the Andes, the Himalayas, or the Alps, the Prince Albert Mountains are being shaped by mountain glaciers. Among them is the David Glacier on the 1,831-meter-high Mt. Joyce. As ice piles on the glacier, it slides under its own weight to the ocean. The ice doesn’t break up when it reaches the ocean; rather, it floats, forming a long tongue of ice. The floating end of the David Glacier is the Drygalski Ice Tongue. These images, acquired by the Moderate Resolution Imaging Spectroradiometer (MODIS) on NASA’s Aqua and Terra satellites, show the ice tongue emerging from the Scott Coast and floating on the Ross Sea with its end pointing towards the upper right corner. This floating spit of ice was recently menaced by the B-15A iceberg, a 120-kilometer-long giant that had been drifting on a collision course with the ice tongue before becoming grounded. The iceberg is visible just below the Drygalski Ice Tongue, mere kilometers away from its end. NASA scientists speculated that if a collision were to occur, the force could break off a section of the ice tongue. No collision was necessary, however; on February 21, 2005, Drygalski calved an iceberg on its own. The five-by-ten-kilometer iceberg was floating away from the ice tongue on February 25, when the second image was acquired. The event is a normal part of the evolution of the ice tongue—pieces regularly break from the tongue as the glacier pushes more ice out over the sea.

<urn:uuid:57e65e90-10b5-47a5-8963-16b44d346088>

# Introduction During the JuliaCon 2023 conference I got several suggestions for writing some more puzzle-solving posts. Therefore this week I want to present a problem that I have recently learned from Paweł Prałat: Assume that you have an even number n of cords of equal length lying in a bunch. They are arranged in such a way that they are approximately straight so that you see one end of each cord in one region (call it left) and the other end in another region (call it right). Imagine, for example, that the cords were wrapped around in their middles. However, this unfortunately means, you cannot distinguish which end belongs to which cord. Your task is to tie the ends of the cords in such a way that after removing the middle wrapping they form a single big loop. Assuming that you tie the cords randomly (left and right ends separately) compute the probability that you are going to succeed. The initial setup of the cords (before we start tying them) is shown on a figure below (I took the image from this source): As usual, we are going to solve it analytically and computationally. The post was written using Julia 1.9.2 and DataFrames.jl 1.6.1. # Analytical solution Denote by p(n) the probability that we succeed with n cords. Notice that we can assume that we can first tie n right ends of the cords in any order. Now let us analyze tying the left ends of the cords. We assume that they are tied randomly. We start tying the left ends by randomly picking cord a and tying it to a random cord b. Let us ask when such a tie is a success and when it is a failure. If n = 2 we know we succeeded. We have just created a loop using two cords. Thus p(2) = 1. If n > 2 what we want to avoid is a situation when the cords a and b are already tied on the right side. Why? Because then we would create a loop that would be smaller than the required loop including all cords. The probability that we pick a wrong end of a cord is 1/(n-1) as we have n-1 ends to choose from and one of them is bad. Thus we succeed with probability (n-2)/(n-1). Now observe that, assuming we succeeded, we have just tied three original cords into a one longer cord. Thus we are left with a situation when we have n-2 cords and the problem has the same structure to what we started with. So we get that for n > 2 we can computep(n) = p(n-2)*(n-2)/(n-1). This means that we can write (using Julia code notation): p(n) = prod((i-1)/i for i in n-1:-2:3) Note that this function works correctly only for even n that is at least 4. We will make its more careful implementation in the computational solution section below. However, first, let us ask ourselves if the formula for this function can be simplified. Indeed we see that it is equivalent to: prod(i-1 for i in n-1:-2:3) / prod(i for i in n-1:-2:3) which in turn can be rewritten as: prod(i-1 for i in n-1:-2:3)^2 / prod(i for i in n-1:-1:2) Now observe that the numerator is just 2^(n-2)*factorial(n÷2-1)^2 (remember that n is even) and the denominator is factorial(n-1). Thus the formula further simplifies to: 2^(n-2)*factorial(n÷2-1)^2 / factorial(n-1) And finally: 2^(n-2) / ((n-1)*binomial(n-2, n÷2-1)) Now from Stirling’s approximation we know that: binomial(n-2, n÷2-1) ~ 2^(2n-4) / sqrt((n-2)*pi) so for sufficiently large n: p(n) ~ sqrt((n/2-1)*pi) / (n-1) Thus we learn that the probability of getting a full circle is of order O(1/sqrt(n)) for large n. Let us now check these results computationally. # Computational solution First start with a more careful implementation of the functions computing p(n): function connect_exact(n::Integer) @assert n > 3 && iseven(n) return prod((i-1)/i for i in n-1:-2:3) end function connect_approx(n::Integer) @assert n > 3 && iseven(n) return sqrt(pi * (n / 2 - 1)) / (n - 1) end Let us check how close the exact and approximate formulas are. Let us compute percentage deviation of the approximation from the exact result: julia> [1 - connect_approx(n) / connect_exact(n) for n in 4:2:28] 13-element Vector{Float64}: 0.11377307454724206 0.060014397013374854 0.040631211300167 0.030689300286045995 0.024649922854770412 0.02059439568578203 0.017683822837349372 0.015493594528168564 0.013785863139806342 0.012417071173843053 0.011295454766000912 0.01035962441429672 0.00956696079055186 We see that approximation is slightly below the exact number and that the percentage deviation decreases as n goes up. With n=28 we are below 1% error. Let us check some larger value of n: julia> 1 - connect_approx(10000) / connect_exact(10000) 2.5004688326446534e-5 We see that the values are now really close. If you were afraid that we might be hitting numeric computation issues with connect_approx since we are multiplying a lot of values, we can easily switch to a more precise computation with Julia: julia> 1 - connect_approx(big(10000)) / connect_exact(big(10000)) 2.500468833607760982625749174941669517305288399515417883351990349709823151295288e-05 We see that using normal Float64 was enough for this range of values of n to get enough accuracy. But what if we were not sure if our derivation of the formula for p(n) was correct? We can use simulation to check it. Here is the implementation of a simulator: function connect_sim(n::Integer) @assert iseven(n) && n > 3 left = randperm(n) neis2 = zeros(Int, n) for i in 1:2:n neis2[left[i]] = left[i+1] neis2[left[i+1]] = left[i] end prev = 1 loc = 2 visited = 2 while true nei1 = isodd(loc) ? loc+1 : loc-1 nei2 = neis2[loc] loc, prev = (prev == nei1 ? nei2 : nei1), loc loc == 1 && return visited == n visited += 1 end end The the code we assume that we numbered the cords from 1 to n and that in the right part they are connected 1-2, 3-4, … (note that we can always re-number them to get this). The neis2 keeps the information about connections on left. To get a random connection pattern we first draw a random n-element permutation and store it in the left variable. Then we assume that the connections are formed by cords left[1]-left[2], left[3]-left[4], … and store these connections in the neis2 vector. Now we are ready to check if this connection pattern is good, that is, it creates one big loop. To do this we start from cord 1 and assume that we first moved to cord 2. The current location of our travel is kept in variable loc. Then from each cord we move either on right or on left to the next cord. The nei1 variable keeps cords neighbor on right and nei2 on left. We keep track in the prev variable which cord we have visited last. Using this information we know which move we should make next. Notice that since we started from 1 we eventually have to reach it. The number of steps taken to reach 1 is tracked by the visited variable. If when loc == 1 we have that visited == n this means that we have formed a big cycle and we return true. Otherwise we return false. Let us check if our simulation indeed returns values close to theoretical ones. For this we will record the mean of 100,000 runs of our simulation (and here the power of Julia shines - it is not a problem to run that many samples). We check the results for the values of n we investigated above: using DataFrames using Random using Statistics connect_sim_mean(n) = mean(connect_sim(n) for _ in 1:100_000) Random.seed!(1234) df = DataFrame(n=[4:2:28; 10_000]) transform(df, :n .=> ByRow.([connect_exact, connect_approx, connect_sim_mean])) The results of running this code are given below: 14×4 DataFrame Row │ n n_connect_exact n_connect_approx n_connect_sim_mean │ Int64 Float64 Float64 Float64 ─────┼────────────────────────────────────────────────────────────── 1 │ 4 0.666667 0.590818 0.66662 2 │ 6 0.533333 0.501326 0.53622 3 │ 8 0.457143 0.438569 0.45843 4 │ 10 0.406349 0.393879 0.40671 5 │ 12 0.369408 0.360302 0.36978 6 │ 14 0.340992 0.33397 0.33996 7 │ 16 0.31826 0.312631 0.31743 8 │ 18 0.299538 0.294897 0.29848 9 │ 20 0.283773 0.279861 0.28399 10 │ 22 0.27026 0.266904 0.26879 11 │ 24 0.25851 0.25559 0.25528 12 │ 26 0.248169 0.245598 0.24755 13 │ 28 0.238978 0.236692 0.24052 14 │ 10000 0.0125335 0.0125331 0.01266 We can see that simulation results match the exact calculations well. # Conclusions I hope you liked the puzzle and the solution. Next week I plan to present the results of some experiments involving machine learning models in Julia.

crawl-data/CC-MAIN-2024-30/segments/1720763514908.1/warc/CC-MAIN-20240719135636-20240719165636-00759.warc.gz

Individual differences | Methods | Statistics | Clinical | Educational | Industrial | Professional items | World psychology | | Eastern Gray Squirrel, Sciurus carolinensis| Eastern Gray Squirrel, Sciurus carolinensis Many, see the article Sciuridae. A squirrel is one of many small or medium-sized rodents in the family Sciuridae. In the English-speaking world, squirrel commonly refers to members of this family's genera Sciurus and Tamiasciurus, which are tree squirrels with large bushy tails, indigenous to Asia, the Americas and Europe. Similar genera are found in Africa. The Sciuridae family also includes flying squirrels, as well as ground squirrels such as the chipmunks, prairie dogs, and woodchucks. Members of the family Anomaluridae are sometimes misleadingly referred to as "scaly-tailed flying squirrels" although they are not closely related to the true squirrels. In United States and Canada, common squirrels include the Fox Squirrel (S. niger); the Western Gray Squirrel (S. griseus); the Douglas Squirrel (Tamiasciurus douglasii); the American Red Squirrel T. hudsonicus; and the Eastern Grey Squirrel (S. carolinensis), of which the "Black Squirrel" is a variant. In Europe the Red Squirrel or Eurasian red squirrel (Sciurus vulgaris) is the most common native species, although the Eastern Grey Squirrel (S. carolinensis) has been introduced in some countries and has displaced the red in many areas, including most of Britain. The word squirrel, first attested in 1327, comes via Anglo-Norman esquirel from the Old French escurel, the reflex of a Latin word sciurus which was itself borrowed from Greek. The word itself comes from the Greek word σκιουρος, skiouros, which means shadow-tailed, because they use their tail to shade their whole body. The native Old English word, 'ācweorna', survived only into Middle English (as aquerna) before being replaced. The Old English word is of Common Germanic origin, with cognates such as German Eichhorn/Eichhörnchen and Norwegian ekorn. Unlike rabbits or deer, squirrels cannot digest cellulose and must rely on foods rich in protein, carbohydrates, and fat. In temperate regions early spring is the hardest time of year for squirrels, since buried nuts begin to sprout and are no longer available for the squirrel to eat, and new food sources have not become available yet. During these times squirrels rely heavily on the buds of trees. Squirrels' diet consists primarily of a wide variety of plant food, including nuts, seeds, conifer cones, fruits, fungi and green vegetation. However some squirrels also consume meat, especially when faced with hunger. Squirrels have been known to eat insects, eggs, small birds, young snakes and smaller rodents. Ground and tree squirrels are typically diurnal, while flying squirrels tend to be nocturnal—except for lactating flying squirrels and their offspring, which have a period of diurnality during the summer. Predatory behavior by various species of ground squirrels, particularly the thirteen-lined ground squirrel, has been noted. Bailey, for example, observed a thirteen-lined ground squirrel preying upon a young chicken. Wistrand reported seeing this same species eating a freshly killed snake. Whitaker examined the stomachs of 139 thirteen-lined ground squirrels, and found bird flesh in four of the specimens and the remains of a short-tailed shrew in one; Bradley, examining white-tailed antelope squirrels' stomachs, found at least 10% of his 609 specimens' stomachs contained some type of vertebrate—mostly lizards and rodents. Morgart (1985) observed a white-tailed antelope squirrel capturing and eating a silky pocket mouse. Relationship with humans Edit Squirrels are generally clever and persistent animals. In residential neighborhoods, they are notorious for eating out of bird feeders, digging in planting pots and flower beds to pull out bulbs which they chew on or to either bury or recover seeds and nuts and for inhabiting sheltered areas including attics and basements. Squirrels use their keen sense of smell to locate buried nuts and can dig extensive holes in the process. Birds, especially crows, often watch a squirrel bury a nut, then dig it up as soon as the squirrel leaves. Although expert climbers, and primarily arboreal, squirrels also thrive in urban environments, where they get used to humans. Their intelligence makes them suitable as pets. As pests Edit Squirrels are sometimes considered pests because of their propensity to chew on various edible and inedible objects. This characteristic trait aids in maintaining sharp teeth, and because their teeth grow continuously, prevents over-growth. Homeowners in areas with a heavy squirrel population must keep attics and basements carefully sealed to prevent property damage caused by nesting squirrels. A squirrel nest is called a "drey". Some homeowners resort to more interesting ways of dealing with this problem, such as collecting and planting fur from pets such as domestic cats and dogs in attics. This fur will indicate to nesting squirrels that a potential predator roams and will encourage evacuation. Fake owls and scarecrows are generally ignored by the animals, and the best way to prevent chewing on an object is to coat it with something to make it undesirable: for instance a soft cloth or chili pepper paste or powder. Squirrel trapping is also practised to remove them from residential areas. However, otherwise squirrels are safe neighbors that pose almost zero risk of transmitting rabies. Squirrels are often the cause of power outages. They can readily climb a power pole and crawl across a power line. The animals will climb onto transformers or capacitors looking for food. If they touch a high voltage conductor and a grounded portion of the device at the same time, they are then electrocuted and cause a short circuit that shuts down equipment. Squirrels have brought down the high-tech NASDAQ stock market twice and were responsible for a spate of power outages at the University of Alabama. To sharpen their teeth they will often chew on tree branches or even the occasional live power line. Rubber plates (squirrel guards) are sometimes used to prevent access to these facilities. Squirrels are blamed for economic losses to homeowners, nut growers, forest managers in addition to damage to electric transmission lines. These losses include direct damage to property, repairs, lost revenue and public relations. While dollar costs of these losses are sometimes calculated for isolated incidents, there is no tracking system to determine the total extent of the losses. Squirrels are also responsible for burrowing into sensitive earthworks such as dams and levees, causing a loss of structural integrity which requires diligent maintenance and prevention. Squirrel burrowing activity has sometimes resulted in catastrophic failures of these structures. As pets Edit Squirrels can be trained to be hand-fed. Because they are able to cache surplus food, they take as much food as is available. Squirrels living in parks and campuses in cities have learned that humans are typically a ready source of food. Urban squirrels have learned to get a lot of food from generous humans. A commonly given food is peanuts, but recent studies show that raw peanuts contain a trypsin inhibitor that prevents the absorption of protein in the intestines. Therefore offering peanuts that have been roasted is the better option. However, wildlife rehabilitators in the field have noted that neither raw nor roasted peanuts nor sunflower seeds are healthy for squirrels, because they are deficient in several essential nutrients. This type of deficiency has been found to cause Metabolic Bone Disease, a somewhat common ailment found in malnourished squirrels. Squirrels are occasionally kept as household pets, provided they are selected young enough and are hand raised in a proper fashion. They can be taught to do tricks, and are said to be as intelligent as dogs in their ability to learn behaviors. Pet squirrels are usually kept without cages, but large cage and a balanced diet with good variety will keep a pet squirrel healthy and happy. The pet owner must beware of "spring fever" at which time a female pet squirrel will become very defensive of her cage, considering it her nest, and will become somewhat aggressive to defend the area. As food Edit Squirrel meat is considered a favored meat in certain regions of the United States where it can be listed as wild game. This is evidenced by extensive recipes for its preparation found in cookbooks, including older copies of The Joy of Cooking. Squirrel meat can be exchanged for rabbit or chicken in recipes, though it can have a gamey taste. Unlike the healthfulness of most game meat, the American Heart Association has found squirrels to be high in cholesterol. In the U.S.Edit In many areas of the U.S., particularly areas of the American South, squirrels are hunted for food. Republican Presidential candidate Mike Huckabee mentioned his experiences eating squirrel during the South Carolina primary, saying that "When I was in college, we used to take a popcorn popper, because that was the only thing they would let us use in the dorm, and we would fry squirrels in a popcorn popper in the dorm room." He later told Meet the Press anchor Tim Russert that squirrel constitutes "a Southern delicacy". The Ramapough Mountain Indian Tribe of New Jesery considered squirrel as an inherent tradition. In the U.K.Edit But in the early 21st century, wild squirrel has become a more popular meat to cook with, showing up in restaurants and shops more often in Britain as a fashionable alternative meat. Specifically, U.K. citizens are cooking with the invasive gray squirrel, which is being praised for its low fat content and the fact that it comes from free range sources. Additionally, the novelty of a meat considered unusual or special has added to the spread of squirrel consumption. Due to the difficulty of a clean kill and other factors, the majority of squirrel eaten in the U.K. is acquired from professional hunters, trappers, and gamekeepers. Some British are eating the gray squirrel as a direct attempt to help the native red squirrel, which has been dwindling since the introduction of the gray squirrel in the 19th century. This factor was marketed by a national "Save Our Squirrels" campaign that used the slogan, “Save a red, eat a gray!” In culture Edit Despite periodic complaints about the animal as a pest, general public opinion towards the animal is favorable, thanks to its agreeable appearance, intelligence and its eating styles and habits. Squirrels are popular characters in many forms of media, such as the literary works of Beatrix Potter, Brian Jacques' Redwall series (including Jess Squirrel and numerous other squirrels), Pattertwig in C. S. Lewis' Prince Caspian, Michael Tod's Woodstock Saga of novels featuring squirrel communities in the style of Watership Down, and the Starwife and her subjects from Robin Jarvis's Deptford novels. Squirrels are also popular characters in cartoons, such as Scrat from Ice Age, Slappy Squirrel of the Animaniacs, Sandy Cheeks from SpongeBob SquarePants, Hammy from Over the Hedge, Benny in The Wild, Rodney from Squirrel Boy, Secret Squirrel, Screwy Squirrel, Nutty from Happy Tree Friends, and Rocky, Bullwinkle's adventuring partner. Grace from the webcomic El Goonish Shive is often pictured as an anthropomorphic squirrel, since it is her most natural and favored form. Video games such as Rare's Conker series starring Conker the Squirrel, as well as Ocean Software's Mr. Nutz. There is even a squirrel-themed super-heroine, Squirrel Girl. Albino squirrels Edit The Albino Squirrel Preservation Society was founded at the University of Texas at Austin in 2001, and its sister chapter at University of North Texas (UNT) petitioned for an election to name their albino squirrel as the university's secondary mascot. The student body narrowly rejected the call. University of Louisville in Kentucky also has a documented population of albino squirrels. Olney, Illinois, known as the "White Squirrel Capital of the World," is home of the world's largest known albino-squirrel colony. These squirrels have the right of way on all streets in the town, and are featured on uniform patches of the local police department. Kenton, Tennessee is home to about 200 albino squirrels. There are also albino squirrels on the main campus of Ohio State University in Columbus, Ohio. Brevard, North Carolina and Marionville, Missouri have a substantial population of white (not albino) squirrels. Western Kentucky University has a locally famous population of white squirrels. Exeter, Ontario in Canada is known for having non-albino white squirrels, believed to be the result of a genetic mutation in the early 20th century. Black Squirrels with white tips on their tails are being noticed throughout Toronto, Ontario. White squirrels are also commonly seen in Dayton, Ohio and on the campus of Youngstown State University in Youngstown, Ohio. The snow belt in Western and Central New York (Buffalo, Rochester, and Syracuse) also has a significant white squirrel population[How to reference and link to summary or text]. Red and grey squirrels in the UK Edit A decline of the red squirrel and the rise of the eastern grey squirrel has been widely remarked upon in British popular culture. It is mostly regarded as the invading greys driving out the native red species. Evidence also shows that grey squirrels are vectors of the Squirrel parapoxvirus for which no vaccine is presently available and which is deadly to red squirrels but does not seem to affect the host. Currently the red squirrel only resides in a few isolated areas of the UK, notably in Scotland, and in England Formby, the Lake District, Brownsea Island, and the Isle of Wight. Special measures are in place to contain and remove any infiltration of grey squirrels into these areas. Under British law, the eastern grey squirrel is regarded as vermin, and at one point it was illegal to release any into the wild; any caught had to be either destroyed or kept captive. In 2008 the law was altered, allowing those with the proper license to release captured grey squirrels. See also Edit - ↑ 1.0 1.1 Squirrel. Online Etymology Dictionary. URL accessed on 2008-02-07. - ↑ Tree Squirrels. The Humane Society of the United States. URL accessed on 2009-01-09. - ↑ Törmälä, Timo, Vuorinen, Hannu; Hokkanen, Heikki (1980). Timing of circadian activity in the flying squirrel in central Finland. Acta Theriologica 25 (32-42): 461–474. - ↑ Friggens, M. (2002). Carnivory on Desert Cottontails by Texas Antelope Ground Squirrels. The Southwestern Naturalist 47 (1): 132–133. - ↑ Bailey, B. (1923). Meat-eating propensities of some rodents of Minnesota. Journal of Mammalogy 4: 129. - ↑ Wistrand, E.H. (1972). Predation on a Snake by Spermophilus tridecemlineatus. American Midland Naturalist 88 (2): 511–512. - ↑ Whitaker, J.O. (1972). Food and external parasites of Spermophilus tridecemlineatus in Vigo County, Indiana. Journal of Mammalogy 53 (3): 644–648. - ↑ Bradley, W. G. (1968). Food habits of the antelope ground squirrel in southern Nevada. Journal Of Mammalogy 49 (1): 14–21. - ↑ Morgart, J.R. (May 1985). Carnivorous behavior by a white-tailed antelope ground squirrel Ammospermophilus leucurus. The Southwestern Naturalist 30 (2): 304–305. - ↑ http://rabies.emedtv.com/rabies/rabies-and-squirrels.html - ↑ K. Muston. Getting Squirrely. Daily Kos:. URL accessed on 2008-02-07. - ↑ Tree Squirrels - University of Georgia - ↑ (2006). Levee Safety Program: Burrowing Animals. Santa Clara Valley Water District. URL accessed on 2008-02-07. - ↑ Jon Gottshall. Jon's World o' Squirrels. Jon's World o' Squirrels. URL accessed on 2007-02-07. - ↑ Susan Saliga. Backyard Squirrel Feeding Tips. Wisconsin Squirrel Connection. URL accessed on 2007-02-07. - ↑ Sara Rowe. Squirrel Tales: Care Instructions For Infant Squirrels. Squirreltales. URL accessed on 2007-02-07. - ↑ http://www.vivavegie.org/BernieandSquirrel.htm - ↑ http://www.boingboing.net/2008/09/11/how-to-make-a-squirr.html - ↑ It tasted like chicken. Retrieved December 19, 2008. - ↑ Davidson, Alan (1999). "Squirrel". Oxford Companion to Food. Oxford University Press. p. 750. ISBN 0192115790. - ↑ Kurlanksy, Mark. The Food of a Younger Land: A Portrait of American Food--Before the National Highway System, Before Chain Restaurants, and Before Frozen Food, when the Nation's Food was Seasonal. Penguin, 2009, p. 112 - ↑ 'Meet the Press' transcript for Feb. 10, 2008. Msnbc.com. Retrieved December 19, 2008. - ↑ http://wcbstv.com/local/ringwood.squirrel.myrtle.2.241671.html - ↑ http://www.9news.com/news/watercooler/article.aspx?storyid=63739 - ↑ 25.0 25.1 25.2 25.3 25.4 25.5 includeonly>Speiler, Marlena. "Saving a Squirrel by Eating One", The New York Times, January 6, 2009. Retrieved on 2009-01-16. - ↑ 26.0 26.1 First, catch your squirrel... - ↑ (Fall 2006). 'Baby' is no more. North Texan 56 (3). - ↑ The Grey/Red Debate. Save our Squirrels. Red Alert North England. URL accessed on 2008-02-07. - ↑ BBC. Virus threatens UK's red squirrels. URL accessed on 2008-05-30. - ↑ Malvern, Jack Captured squirrels live to nibble again. The Times. URL accessed on 2009-06-07. - The Scholarly Squirrel, general information about squirrels - Tree of Life: Sciuridae - About White Squirrels - California squirrels defend themselves - National Geograhphic link on Squirrels |This page uses Creative Commons Licensed content from Wikipedia (view authors).|

<urn:uuid:7cc4013d-86eb-4c07-af8b-3420aa705223>

You are here Circle game activities Conditional chain game This game is good to revise and practise structures in the first conditional. - The teacher begins with a sentence, for example 'If I go out tonight, I'll go to the cinema.' - The next person in the circle must use the end of the previous sentence to begin their own sentence. E.g. 'If I go to the cinema, I'll watch The Last Samurai.' The next person could say, 'If I watch The Last Samurai, I'll see Tom Cruise,' etc. etc. A very simple game where students must think of words connected to the word that comes before. - For example, the teacher says, 'fish', the next person thinks of a word they associate with fish, such as 'water', the next person could say 'a glass', the next 'window', etc. - You can decide as a group if associations are valid. Ask the student to justify the connection. - To make it more competitive, set a thinking time limit and eliminate students. - When they are eliminated they can become judges. Chinese whispers - telephone lines A sentence is whispered around the circle. The last student to receive the message either says it aloud or writes it on the board. This can be a fun way to introduce a topic and activate schema at the beginning of a class. For example, for a class on food, whisper the question, 'What did you have for lunch today?' Equally, at the end of a class it can be a nice way to revise structures or vocabulary from the lesson. - To begin with, students sit in a circle and do the hand actions of lap (both hands to lap), clap, left click, right click. - When they get the hang of it, add these words in time to the rhythm 'Concentration, concentration, concentration now beginning, are you ready? If so, let's go!' - On the first finger click, you say your name, and on the second click you say the name of someone in the circle. - You have passed the turn to the person you nominated on your second finger click. - Then they say their own name on the first click and the name of another student on the second. - When they have got the idea, use lexical sets. For example, everyone says their favourite sport first then use these to play the game. - For a competitive group, eliminate those students who make mistakes. I went to the shops and I bought… The classic memory game where each person adds a new item to the list in alphabetical order. - For example, student 1, 'I went to the shops and I bought an apple'. Student 2, 'I went to the shops and I bought an apple and a bike'. Student 3, 'I went to the shops and I bought an apple, a bike and a coat'. - This game can be adapted to different levels and lexical sets. I recently revised sports and the use of do / play / go by playing 'I went to the sports centre…', the same game but using different vocabulary. For example, 'I went to the sports centre and I did aerobics', 'I went to the sports centre and I did aerobics and played basketball', 'I went to the sports centre and I did aerobics, played basketball and went canoeing' etc. Yes / No game - Nominate one student to be in the hot seat, slightly apart from the rest of the circle. - The rest of the group must think of questions to ask the student in the hot seat. - They can ask anything they like, the only rule is that the student in the hot seat must answer the questions without using the words 'yes' or 'no'. - Also ban 'yeah', head nods and shakes! For example, a student asks, 'Are you wearing jeans today?' The student in the hot seat could reply, 'I am' or 'You can see that they're jeans!' For advice on how to manage these games and for more activity ideas, see:

<urn:uuid:4581ddd3-011a-425b-8ce5-ca809caf0496>

Room 3 -Ships - the 15th century There are two differences between boats and ships - boats are smaller than ships and boats can be powered by one or more people pulling or pushing the boat through the water. Ships are larger and need to be powered by more than manpower. The first ships came to America with the Europeans. Europeans had been using ships for many centuries. These ships were powered by wind that was captured in material made into sails. Large ships were built using many sails so that ships could go faster and could carry more people and things. Activity - Discover how wind can power sails. Teacher will give each student a tissue. Students open tissues and hold on to sides of the tissue. They walk around room with tissue opened. Then they blow in tissue to simulate wind and compare - did the air in the room make the tissue move? did the wind (from blowing) make the tissue move? Which made the tissue move more? Visit some sailing ships at Mystic Seaport. Look at the globe. Why would people need to have sails and not depend on manpower pulling or pushing oars or paddles to power a boat from Europe to North America? Find out what it would be like to be a sailor. What jobs needed to be done? Who were the people who sailed on a sailing ship? As time went on, people wanted to sail farther and faster. In America in 1787 John Fitch made the first steamship. Wood was burned to make steam which turned a wheel which made a boat go. Today when we think of steamships we usually think of Robert Fulton who made the first commercially successful steamship in 1807. As time went on, people found the paddlewheel steamer too slow. Also it could not be used on all types of water. Steam engines were improved. Propellors were used instead of paddlewheels to move the boat. People still wanted to go farther and faster. They added more engines to the steam ships. Coal was burned instead of wood. It was easier to store a lot of coal than to store a lot of wood. Today ships use oil instead of coal to power the engines. Activity - Look at these pictures of some steamships. Can you tell if they have more than one engine? What helps you to know? Find out where naval and maritime museums are in your area. GO TO HOME PAGE.

<urn:uuid:85c4fb16-2dd3-4243-b057-d5634cd6266b>

# What You Need to Know About What Is Polynomial in Mathematics and Why ## The What Is Polynomial in Mathematics Trap A polynomial with one term is referred to as a monomial. A plain number may also be a polynomial term. This will give me the top term, the maximum power of x. The quadratic is the perfect square. You are able to imagine that we're now graphing a lot of 32-dimensional clouds in a bigger space. If we were attempting to address an equation of this polynomial set equal to zero, we'd need to use the quadratic formula if we didn't understand how to finish the square. The very first term in a polynomial is known as a top term. By writing polynomials in descending powers it's possible to find out what the level of the polynomial is by simply studying the very first term. academic essays online They are just one kind of algebraic expression. Also, it's the only one with an endless number of roots. To locate the minimum of these problems the derivative can nevertheless be used, but we will need to use partial derivativesone with respect to every coefficient. When you have reduced the polynomial to a quadratic, you may use the quadratic formula to get the remaining two roots. ## The Argument About What Is Polynomial in Mathematics There are not any higher terms (such as x3 or abc5). Their response wasn't restricted to a single reason. The solution to this question is the very first term of the quotient. This conversion of a single problem into another is known as reduction. You will learn the way to select your factors and check your solution. If you prefer something actually helpful for pen-and-paper solutions, you may want to know the true theory supporting the solution. ## What Is Polynomial in Mathematics Features Suppose you're assigned to compose a program. http://www.ciencia.gob.es/portal/site/MICINN/menuitem.4a5b2889f2e6e245c1e45ddc026041a0/?vgnextoid=57517fd156195510VgnVCM1000001d04140aRCRD You must monitor the process at the important control factor and maintain records to demonstrate that the vital restrictions are happy. Typically this procedure is done multiple times till you have identified all the solutions. This done by means of a process referred to as analytical continuation, leading to a unique generalized function. There are various sorts of polynomial functions dependent on the level of the polynomial. Polynomial functions are rather simple to comprehend. ## The Hidden Gem of What Is Polynomial in Mathematics Additional risk models should be trained on a lot of information. So the only means to improve accuracy is to raise the true-positives www.grademiners.com/coursework-writing and the true-negatives. Now what's a polynomial it's really merely a finite amount of power functions. The summations will become very large. Polynomials are extremely essential to understand to be effective in algebra. They are used in a wide variety of problems where they are called as polynomial equations. The learned hyperplane is dependent on equation 1. Much like fractions, the numerator of the aforementioned expression is known as the dividendand the denominator is known as the divisor. It is known as a fifth degree polynomial. I believe I still have to tweak to acquire much better radius of curvature estimates. Within the next section we'll learn more about the graphs of polynomials. Within this section we'll learn more about the neighborhood behavior of polynomials generally speaking. In addition, XCode did a very very good job at spoiling me by auto-completing a good deal of typing. The brief answer isn't that I know of. If you follow all the steps in the former list, you'll have a simple time with factoring trinomials. ## What Is Polynomial in Mathematics Fundamentals Explained Some courses will nonetheless need you to buy a text book. We are aware that mathematics is definitely useful, however, it's so difficult to explain to students how it can be helpful for them. Other such techniques exist too. This is particularly important if the GCF contains a variable. We'll implement a very simple kind of Gradient Descent using python. Confirm the division algorithm. ## If You Read Nothing Else Today, Read This Report on What Is Polynomial in Mathematics There's a means to tell, and there are a couple calculations to do, but it's all easy arithmetic. Prediction won't be as good in areas of the data which were not present in the training data. Try to remember, just 32 parameters describe the majority of the observable universe. Yeas, it can be better with a great deal of data, but nevertheless, it may also be good with merely a few data points. This is beneficial for prediction. So, the sum of all of the residuals is the expected price of the residuals times the entire number of information points. ## New Ideas Into What Is Polynomial in Mathematics Never Before Revealed Suppose our dataset includes measurements of the range of three distinct fruits that have the properties of being long and sweet, as given in the next table. The property of symmetry could be illustrated by the next example. The overall model is shown in the next figure. In our instance, the outcome is 8x3 8x2. Be cautious of the signs when you do so. It shows the amount of terms in a comprehensive polynomial of a particular order and the variety of terms connected with a specific purchase. #### Colabora Propiedad de las fotografías: Ayto. de Monzón, Carlos Orteu, Jesús Ginestra, Lúa Media.

crawl-data/CC-MAIN-2020-05/segments/1579250593994.14/warc/CC-MAIN-20200118221909-20200119005909-00523.warc.gz

# Question #16ef7 May 3, 2016 Fireworks should explode after $6.25$ seconds. #### Explanation: We seek the point (here time $t$) where the quadratic function $s \left(t\right)$ reaches its maximum value. Fortunately we have a simple formula for that: A quadratic function $s \left(t\right) = a {t}^{2} + b t + c , \setminus \setminus a \ne 0$ reaches its maximum value (if $a < 0$; if $a > 0$ this function has only a global minimum but the following formula holds) at ${t}_{\max} = - \frac{b}{2 a}$. If we're also interested in the value at this point we can simply evaluate it: $s \left({t}_{\max}\right)$. In our particular fireworks' case we have: ${t}_{\max} = - \frac{200}{2 \cdot \left(- 16\right)} = \frac{25}{4} = 6.25 \setminus \setminus \left[s\right]$ Hence, fireworks should explode after $6.25$ seconds. If the question was 'What height the fireworks will explode at?' we would also have to evaluate the maximum value: $s \left({t}_{\max}\right) = - 16 \cdot {6.25}^{2} + 200 \cdot 6.25 + 4 = 629 \setminus \setminus \left[m\right]$. May 3, 2016 $6.25$ seconds. #### Explanation: You need to find the height at which the fireworks stop rising and start falling, which you can do by differentiating the equation to find the maximum (the point at which the gradient is zero). $s ' \left(t\right) = - 32 t + 200$ must equal zero at the highest point. $32 t = 200$ $t = \frac{200}{32} = 6.25$ secs.

crawl-data/CC-MAIN-2021-39/segments/1631780057158.19/warc/CC-MAIN-20210921041059-20210921071059-00385.warc.gz

# PYTHAGORAS THEOREM!!!QUESTION 1: For what values of n does {n, n + 1, n + 2} form a pythagorean triple. QUESTION 2: Show that {n, n + 1, n + 3} cannot form a pythagorean triple. krishna-agrawala | College Teacher | (Level 3) Valedictorian Posted on Pythagoras theorem states that in a right angled triangle the square of hypotenuse equals sum squares of other two sides. A Pythagoras triple refers to a set of three positive integers a,b, and c that satisfy the condition: a^2 + b^2 = c^2 To establish that a set of three integers forms a Pythagoras Triple we have to prove that the sum of squares of the two smaller integers is equal to the square of the square of the third number. Question 1 To find the value of n for which the set of numbers represented by (n, n+1, n+2) form a Pythagoras triple we form a equation based on on the condition of Pythagoras triple and then solve the equation for value of n. Thus: n^ + (n+1)^2 = (n+2)^2 n^ + (n+1)^2 - (n+2)^2 = 0 n^ + n^2 + 2n + 1- n^2 - 4n - 4 = 0 n^ - 2n - 3 = 0 n^ - 3n + n - 3 = 0 n(n - 3) + 1(n - 3) = 0 (n + 1)(n - 3) = 0 Therefore n = 3 0r n = -1 Thus the condition of Pythagoras triple is satisfied for n = 3 Question 2 To prove that the set of numbers represented by (n, n+1, n+3) cannot form a Pythagoras triple we form a equation based on on the condition of Pythagoras triple,  solve the equation for value of n, and then show that these values of n are not integers Thus: n^ + (n+1)^2 = (n+3)^2 n^ + (n+1)^2 - (n+3)^2 = 0 n^ + n^2 + 2n +1 - n^2 - 6n - 9 = 0 n^2 - 4n - 8 = 0 n^2 - 4n + 4 = 12 (n - 2)^2 = 12 n - 2 = 12^1/2 = 3.4641 n = 3.4641 + 2 = 5.4641 As only possible value of n is not an integer, the given set of number cannot form a Pythagoras triple. neela | High School Teacher | (Level 3) Valedictorian Posted on Obviously the greatest side is the htpotenuse. So n+2 should be the sum of the squares on the other two sodes , n+1 and n forming the right angle. So, (n+2)^2 = (n+1)^2+n^2. Or (n^2+4n+4) = (n^2+2n+1)+n^2 . Or 0 = 2n2+2n+1 - (n^2+4n+4) . Or 0 = n^2-2n -3. Or 0 = (n+1)(n-3). Or n-3 = 0 gives n =3. Therefore, 3 , 3+1= 4 and 3+2=5. 2) Let us assume that n^2+(n+1)^2 =  (n+3)^2. Or n^2 +n^2+2n+1= n^2+6n+9. Or 2n^2+2n- (n^2+6n ) = 9-1. Or n^2-4n = 8 Or n^2 -4n + 4 = 8+4 = 12 . Or (n-2)^2 = sqrt12. Or n = 2+2*3^(1/2). Therefore , 2+2*3^(1/2), 3+2*3^(1/2)  and 5+2*3(1/2) are  the pytagorus triple, with the charecter, n, n+1 and n+3, but they are not the integers.

crawl-data/CC-MAIN-2017-04/segments/1484560281450.93/warc/CC-MAIN-20170116095121-00512-ip-10-171-10-70.ec2.internal.warc.gz

# 44°C To Fahrenheit: Conversion Formula, Common Temperatures, And Benefits // Thomas Discover the conversion formula and practical applications for converting 44°C to Fahrenheit. Compare common temperatures and understand the of knowing the conversion. ## What is 44°C in Fahrenheit? ### Understanding the Conversion Formula Converting Celsius to Fahrenheit is a common task when dealing with temperature measurements. If you have come across the temperature 44°C and want to know its equivalent in Fahrenheit, we can help you with that. Understanding the is the key to making accurate conversions. To convert Celsius to Fahrenheit, you can use the following formula: ``°F = (°C × 9/5) + 32`` Let’s break down the formula to gain a better understanding. The first step is to multiply the Celsius temperature by 9/5. This accounts for the difference in the scales of Celsius and Fahrenheit. Then, you add 32 to the result to obtain the temperature in Fahrenheit. For example, let’s apply this formula to convert 44°C to Fahrenheit: ``````°F = (44 × 9/5) + 32 = (396/5) + 32 = 79.2 + 32 = 111.2°F`````` Therefore, 44°C is equal to 111.2°F. Understanding the conversion formula allows you to convert Celsius to Fahrenheit accurately. This knowledge can be valuable in various situations, from cooking recipes that use Fahrenheit measurements to understanding weather forecasts in different regions. Now that you know the conversion formula, let’s explore the step-by-step process of converting 44°C to Fahrenheit in the next section. ## How to Convert 44°C to Fahrenheit Converting Celsius to Fahrenheit may seem intimidating at first, but it’s actually a straightforward process. By following the step-by-step conversion process below, you’ll be able to easily convert 44°C to Fahrenheit. ### Step-by-Step Conversion Process To convert 44°C to Fahrenheit, you can use the simple conversion formula. Just follow these steps: 1. Start by multiplying the Celsius temperature by 9/5 (or 1.8). 2. In this case, you would multiply 44 by 1.8, which equals 79.2. 3. Next, add 32 to the result from step 1. 4. Adding 32 to 79.2 gives us a final result of 111.2. So, when you convert 44°C to Fahrenheit using the step-by-step process, you get 111.2°F. It’s important to note that the formula for converting Celsius to Fahrenheit is a standard formula that applies to any Celsius temperature. By understanding this simple process, you can easily convert Celsius temperatures to Fahrenheit and vice versa. Remember, 44°C is the equivalent of 111.2°F in Fahrenheit. This conversion can be useful in various scenarios, such as when traveling to a country that uses the Fahrenheit scale or when reading temperature measurements in different units. ## Why Convert Celsius to Fahrenheit? ### Differences in Temperature Scales Have you ever wondered why there are different temperature scales used around the world? The Celsius and Fahrenheit scales are two of the most widely used temperature measurement systems. Understanding the differences between these scales can help us appreciate the need to convert temperatures from Celsius to Fahrenheit and vice versa. The Celsius scale, also known as the centigrade scale, is based on the freezing point and boiling point of water. On this scale, water freezes at 0 degrees Celsius and boils at 100 degrees Celsius at sea level. The Celsius scale is commonly used in most countries around the world, making it the standard for scientific and everyday temperature measurements. On the other hand, the Fahrenheit scale, named after the physicist Daniel Gabriel Fahrenheit, was developed in the early 18th century. Unlike the Celsius scale, the Fahrenheit scale has its zero point set at the lowest temperature Fahrenheit could achieve using a mixture of ice and salt. The freezing point of water is set at 32 degrees Fahrenheit, while the boiling point of water is 212 degrees Fahrenheit at sea level. One key difference between the Celsius and Fahrenheit scales is the size of the degree increments. In the Celsius scale, each degree represents a smaller change in temperature compared to the Fahrenheit scale. This means that a one-degree change in Celsius is equivalent to a larger change in Fahrenheit. As a result, when converting temperatures between these two scales, the values often appear to be significantly different. Another difference lies in the range of temperatures covered by each scale. The Celsius scale spans from the freezing point of water to its boiling point, making it ideal for everyday temperature measurements. On the other hand, the Fahrenheit scale has a wider range, extending from extreme cold temperatures to high heat. This makes it particularly useful in certain industries, such as weather forecasting and engineering. Understanding the differences between the Celsius and Fahrenheit scales is essential for accurately interpreting and comparing temperatures. Whether you’re planning a trip abroad, studying science, or simply curious about temperature conversions, knowing how to convert Celsius to Fahrenheit and vice versa can be incredibly beneficial. In the next section, we will delve into the step-by-step process of converting 44°C to Fahrenheit, providing you with the tools to navigate between these two temperature scales effortlessly. ## Common Temperatures in Celsius and Fahrenheit When it comes to measuring temperature, there are two common scales used worldwide: Celsius and Fahrenheit. Understanding the relationship between these scales can help us make sense of temperature readings and comparisons. Let’s take a closer look at the Celsius and Fahrenheit scales and how they relate to each other. ### Comparing 44°C to Other Values To better understand the temperature of 44°C, let’s compare it to some other values on the Fahrenheit scale: • 44°C is equivalent to 111.2°F. This means that if the temperature is 44°C, it would be considered quite hot in Fahrenheit. • In comparison, a temperature of 0°C (freezing point of water) is equivalent to 32°F. So, 44°C is significantly higher than the freezing point of water in Fahrenheit. • On the Fahrenheit scale, 44°C is closer to 100°F, which is the typical body temperature of a healthy human being. So, if the temperature outside is 44°C, it would be similar to having a fever in Fahrenheit. • To put it into perspective, let’s consider some other common temperatures. Room temperature, which is typically around 20-25°C, is equivalent to 68-77°F. So, 44°C is much higher than room temperature in Fahrenheit. • In extremely hot climates, such as deserts, temperatures can reach above 50°C (122°F). So, while 44°C may be considered hot, it is not extreme when compared to some other regions around the world. By comparing 44°C to other values on the Fahrenheit scale, we can get a better sense of its significance and understand how it relates to temperatures we are more familiar with. Whether it’s for weather forecasting, cooking, or simply understanding the temperature outside, knowing the conversion between Celsius and Fahrenheit can be helpful in various practical scenarios. ## Benefits of Knowing the Conversion ### Practical Applications and Scenarios Knowing how to convert Celsius to Fahrenheit can be incredibly beneficial in various practical applications and scenarios. Whether you are traveling to a country that uses the Fahrenheit scale or need to understand temperature readings in different units for work or personal reasons, having this knowledge can prove to be very useful. Here are some specific scenarios where understanding the Celsius to Fahrenheit conversion can come in handy: 1. Traveling and Weather: When traveling to countries that use the Fahrenheit scale, knowing how to convert Celsius to Fahrenheit allows you to understand and interpret local weather forecasts accurately. It helps you plan your activities and make appropriate clothing choices based on the temperature readings. 2. Cooking and Baking: Many recipes, especially those from different countries, provide temperature instructions in Celsius or Fahrenheit. Being able to convert between the two scales ensures that you can follow the recipe correctly and achieve the desired results in your culinary endeavors. 3. Medical and Health: In some medical contexts, temperature measurements may be presented in Celsius or Fahrenheit. Being able to convert between the two units can help you understand and communicate temperature readings effectively, whether it’s for personal health monitoring or discussing medical information with healthcare professionals. 4. Scientific Research: Scientific research often involves analyzing temperature data collected in different units. Understanding the conversion between Celsius and Fahrenheit enables researchers to compare and interpret temperature measurements accurately, ensuring the validity and reliability of their findings. 5. International Business: In international business settings, temperature-sensitive industries such as agriculture, food, and pharmaceuticals often require temperature conversions. Having the ability to convert between Celsius and Fahrenheit allows for seamless communication and collaboration across borders, facilitating efficient decision-making and problem-solving. 6. Educational Purposes: Teaching and learning about temperature and temperature conversions are fundamental aspects of science education. Being able to explain the conversion process and its practical applications helps students grasp the concept better and apply it in real-world scenarios. By knowing the conversion between Celsius and Fahrenheit, you open yourself up to a range of opportunities and enhance your understanding of temperature measurements in different contexts. It empowers you to make informed decisions, communicate effectively, and navigate various situations where temperature plays a crucial role. In the next section, we will delve into the differences between Celsius and Fahrenheit temperature scales to further understand why these conversions are necessary. ### What is the Formula for Converting Celsius to Fahrenheit? Converting Celsius to Fahrenheit requires the use of a simple formula. To convert a temperature from Celsius to Fahrenheit, you can use the following formula: F = (C × 9/5) + 32 In this formula, F represents the temperature in Fahrenheit, and C represents the temperature in Celsius. By plugging the value of 44°C into this formula, you can easily calculate the equivalent temperature in Fahrenheit. ### Is 44°C Considered Hot or Cold in Fahrenheit? When we convert 44°C to Fahrenheit, we find that it is equivalent to approximately 111.2°F. In terms of Fahrenheit, this temperature is considered quite warm. It falls within the range of temperatures that are generally associated with hot weather. However, the perception of hot or cold can vary depending on individual preferences and acclimation to different climates. ### How Accurate is the Conversion Formula? The conversion formula for Celsius to Fahrenheit is a widely accepted and accurate method for converting temperatures between the two scales. The formula provides an accurate approximation of the temperature conversion, ensuring that the converted value is reliable for practical purposes. However, it’s important to note that the formula is based on mathematical calculations and may have slight variations due to rounding or other factors. ### Can I Use an Online Converter for 44°C to Fahrenheit? Yes, using an online converter is a convenient and efficient way to convert 44°C to Fahrenheit. Numerous websites and mobile applications offer easy-to-use conversion tools that instantly provide the converted temperature. These online converters utilize the same conversion formula mentioned earlier, ensuring accurate and reliable results. Using an online converter is particularly helpful when you need to convert multiple temperatures quickly or when you don’t have immediate access to a calculator. Contact 3418 Emily Drive Charlotte, SC 28217 +1 803-820-9654

crawl-data/CC-MAIN-2024-18/segments/1712296817106.73/warc/CC-MAIN-20240416191221-20240416221221-00724.warc.gz

There was a problem with two flue irons; they could cut their way back out, and a whale would be lost. When the whale ran and force was applied to the whale line, the harpoon shaft bent, and in doing so the two flue head could be bent and repositioned such that only one flue caught in the flesh to hold fast. The opposite flue would then be positioned with the sharp edge presented to uncut flesh. The force on the whale line and shaft, plus the motions imparted by the fleeing whale, caused the sharp flue to cut its way out. With one flue removed, only the single flue would catch and hold fast. The broad flat side of the head would be presented to uncut flesh rather than the sharp edge of the opposite flue. Another difficulty with two flue irons was the size of the head across the flue tips. Blubber was difficult to penetrate; the larger the harpoon head, the more difficult it was to dart the iron deep enough to hold fast. A simple remedy was to eliminate one flue to reduce the width of the head. The single flue iron was developed sometime around the early 1820's. No one in particular is given credit for this design. The first recorded use in the American fishery occurred in 1824. Reuben Delano, in Wanderings and Adventures of Reuben Delano, Being a Narrative of Twelve Years Life in a Whale Ship!, Boston, 1846, wrote about his voyage in Ship Stanton of Fairhaven, 1824 - 1827. Early in the voyage in 1824: Our boats were cleared away, and our first officer was soon "on and fast" to a good sized whale, with a one flud [sic] iron which did not hold him in tow fifteen minutes before it drew. There was not much variation to the single flue design. Some flues were long and curved while others were short and hefty. This harpoon became popular about 1840, but never completely replaced the two flue irons. The single flue iron was short lived due to the development of the toggle iron in 1848, but even so it was preferred by some and was found in a whaleship's inventory through the 19th century, although in decreasing numbers. (See the tabulation for harpoon inventory over the years for a typical whaleship on the Harpoon page). ©: 2000 - 2008Thomas G. Lytle . All rights reserved Back to Main Menu Bibliography Harpoons History Knives Lances Links Markings Non-Whaling Patents Poison Processing Shoulder-Guns Shoulder-gun Irons Spades Swivel Guns and Irons

<urn:uuid:beea7e0e-f6b9-4115-9832-752f12257b12>

# The 4 4s problem… v100 Once again, credits to Enigmatic Code for providing the post. Puzzle 55: Ton up How can you divide 100 into four parts such that: adding 4 to the first part, subtracting 4 from the second part, multiplying the third part by 4 and dividing the fourth part by 4 results in all parts having the same value as each other? https://enigmaticcode.wordpress.com/2020/04/18/puzzle-55-ton-up/ Alright… This was extremely simple to solve, and I solved it in about 30 seconds. Define the same value at the end of the puzzle to be “a”. Then, the four values can be represented as: • Adding: a – 4 (since adding 4 to a – 4 gives a) • Subtracting: a + 4 • Multiplying: a/4 • Dividing: 4a These four parts all add up to 100, so a – 4 + a + 4 + a/4 + 4a = 6a + a/4, or 25a/4 = 100. Multiply each side by 4, and 25a = 400. Therefore, a = 16. The answer to the problem, however, isn’t the end result; it’s the beginning values. That’s simple: • a – 4 = 12 • a + 4 = 20 • a/4 = 4 • 4a = 64 To double-check, 12+20+4+64 = 100. It works! It’s amazing how things could seem extremely difficult on the outside, yet the solution is such a simple one. Anyways, see ya in the next post! This site uses Akismet to reduce spam. Learn how your comment data is processed.

crawl-data/CC-MAIN-2023-14/segments/1679296945376.29/warc/CC-MAIN-20230325222822-20230326012822-00053.warc.gz

Pennsylvania, like many other colonies, was involved with the war and developing problem of the American Revolution. Philadelphia was the site of the Constitutional Convention. It also served as the]] for a while. Threats from Britain caused the country to relocate to Baltimore, Maryland. They returned to Philadelphia later and moved on later. The Pennsylvania Line was a formation within the Continental Army, composed of infantry regiments from the state of Pennsylvania. A troop commanded by General "Mad" Anthony Wayne was surprised by a bayonet charge at Paoli, Pennsylvania. The massacre claimed more than 50 Americans and more than 100 were wounded. Gen. Washington kept his forces at Valley Forge near the Schuylkill River. - Bibliography of the Continental Army in Pennsylvania compiled by the United States Army Center of Military History |This page uses Creative Commons Licensed content from Wikipedia (view authors).|

<urn:uuid:dc7edd20-993b-4a0f-97c1-6a8666bb8368>

A satellite orbiting the sun has identified its 2,000th comet after 15 years in space. But the Solar and Heliospheric Observatory, known as SOHO, wasn't designed as a comet-hunting spacecraft; its primary research goal is to study the outer atmosphere of the sun. A satellite with a primary mission to study the sun has proven to be a fortuitous comet hunter. The Solar and Heliospheric Observatory, known as SOHO, recently identified its 2,000th comet. Launched on Dec. 2, 1995, SOHO's main research goal is to study the outer atmosphere of the sun, known as the corona. In the process of orbiting and imaging the sun, SOHO blocks out the brightest part and sends back pictures that are ideal for comet hunters to pore over. Amateur and professional astronomers alike are on the lookout for spots of the right brightness that are headed toward the sun -- characteristics of comets. On Tuesday, NASA announced that SOHO had successfully identified its 2,000th comet on Dec. 26. The find was made by Michal Kusiak, an astronomy student at Jagiellonian University in Poland. Kusiak also found the 1,999th comet and more than 100 other comets since 2007. It took astronomers 10 years to identify the first thousand comets, but just five years to find the next 1,000. NASA says this partly because more people are looking at the images, but also because of an unexplained systematic increase in the number of comets around the sun. Copyright 2010 National Public Radio. To see more, visit http://www.npr.org/.

<urn:uuid:dfdbadf9-a5c4-4d27-9b8e-67188a3c780f>

# Doubling Money Game The casino offers a certain win-lose game, where you have $p$ chance of winning. You can bet any amount of money, and if you win you get twice your bet; otherwise, you lose your bet. If you use the optimal strategy, what is your chance of doubling your money, as a function of $p$? I came up with the following incorrect solution: You have a 100% chance of winning if $p>\frac{1}{2}$ and $p$ chance of winning if $p<\frac{1}{2}$. Suppose that $p>\frac{1}{2}$. Then each time you bet exactly half your money. If you have $x$ dollars, you end up with $\frac{3}{2}x$ if you win and $\frac{1}{2}x$ if you lose, hence your expected outcome is $\frac{3}{2} xp + \frac{1}{2}x(1-p)$ which equals $xp + \frac{1}{2}x$. So if $p>\frac{1}{2}$, then the total is greater than $x$. Given that each game on average gains you money and you can play an arbitrary number of games, of course you should have 100% chance of doubling your money. Similarly, if $p<\frac{1}{2}$, it can be shown that no matter how much we bet, we lose money on average. Then on average our money will tend to go toward zero, so we're better off just going all-in at the start, with $p$ chance of doubling our money. I do not understand the proper solution, but I think this solution is incorrect; however, I'm having trouble pinpointing where my proof falls apart. Thanks. Edit: for reference I've included a screenshot of the given solution. • What makes you think this is incorrect? – Ross Millikan Sep 29 '11 at 23:58 • Looks good to me, too. (Of course the casino may not allow you to bet fractional dollars, so perhaps a more realistic strategy for the $p>1/2$ case would be to bet one chip at a time until you've either doubled your initial money or gone broke.) – Henning Makholm Sep 30 '11 at 0:24 • @gambel An interesting problem! What book is this from? – user940 Sep 30 '11 at 12:22 • It's from the Wohascum County Problem Book, and it's supposedly one of the harder problems. – gambel Sep 30 '11 at 14:03 This answer only examines the betting strategy the OP advocates (always bet half of what you have) and some of its variants. With this strategy: One never gets broke (one's fortune is always positive). One doubles up almost surely if $p$ is large enough but with a probability which is less than $1$ if $p$ is small enough. Note that there is no theorem saying that every positive submartingale reaches every level almost surely, hence arguments based on averages of increments only, cannot suffice to reach a conclusion. In the strategy the OP advocates, the fortune performs a multiplicative random walk whose steps from $x$ are to go to $\frac32x$ and to $\frac12x$ with probability $p$ and $1-p$ respectively. Thus, the logarithm of the fortune performs a usual random walk with steps $\log\frac32$ and $\log\frac12$, whose constant drift is $m(p)=p\log\frac32+(1-p)\log\frac12$ hence $m(p)=p\log3-\log2$ has the sign of $p-p^*$ where $p^*$ solves the equation $3^{p^*}=2$, hence $p^*=\frac{\log2}{\log3}=.6309...$ If $p\geqslant p^*$, $m(p)$ is nonnegative hence the logarithm of the fortune reaches the set $[C,+\infty)$ almost surely for every $C$. In particular, one doubles up almost surely. If $p<p^*$, $m(p)$ is negative hence the logarithm of the fortune has a positive probability $q$ to never visit the set $[\log2,+\infty)$, for example. This means that with probability $q$, one will bet an infinite number of times without ever getting broke nor doubling up. In effect the fortune at time $k$ will go to zero like $a(p)^k$ when $k\to+\infty$, with $a(p)=\frac123^p<1$. There is no easy formula for the probability $q$ to never double up but the probability to double up $n$ times decreases exponentially like $\exp(-b(p)n)$ when $n\to\infty$, where $b(p)$ is the unique positive solution of the equation $p(3^b-1)=2^b-1$. If $p<p^*$, the strategy where one always bets half of what one has fails. What happens with the strategy where one always bets a proportion $r$ of what one has? The same analysis applies and shows that one doubles up almost surely, for every $p\geqslant\wp(r)$, where $p=\wp(r)$ solves the equation $(1+r)^{p}(1-r)^{1-p}=1$ (note that $\wp(\frac12)=p^*$). The other way round, for every given $p>\frac12$, a strategy which wins almost surely is to bet a proportion $r$ of what one has, provided $p\geqslant\wp(r)$. Since $\wp(r)\searrow\frac12$ when $r\searrow0$, one gets: For each $p>\frac12$, the strategy where one always bets a proportion $r$ of what one has wins almost surely for every small enough positive value of $r$ (for example, $r\leqslant 2p-1$). • Ah that seems to be right, it would indeed make sense to use the logarithmic scale if we are multiplying and dividing. – gambel Sep 30 '11 at 22:54 I will assume that you know about measure theory as the basis of probability theory. I'll first try to provide a precise formulation of the problem for $p \gt \frac{1}{2}$ and then try to explain what is missing in your explanation. We have as probability space the space of bets $$\Omega = \{-1, 1 \}^{\mathbb{N}}$$ where the event 1 denotes "win" and -1 denotes "loose", and the probability of $1$ is $p$. We have an initial budget of $z_0$ dollars. We can bet arbitrarily small amounts of money. A betting strategy would be a sequence $(a_i)$ denoting the amout of money $\geq 0$ you bet, $$(a_i) \in \mathbb{R_+^\mathbb{N}}$$ such that for all $n \in \mathbb{N}$ $$\sum_{i=1}^n \omega_i a_i \ge -z_0$$ This condition says that you cannot bet money that you don't have, i.e. once you have lost $z_0$ you have to stop (and bet $0$ dollars from that moment for all following bets). When you write you should have 100% chance of doubling your money. your claim is that for $p \gt \frac{1}{2}$ there is a betting strategy $(a)_i$ so that $$\text{winning condition: }P(\omega: \text{there is an index n such that } \sum_{i=1}^n \omega_i a_i \ge z_0) = 1$$ Your betting strategy to back up your claim is each time you bet exactly half your money. Let's call this betting strategy $(b)_i$. You'd have to prove that the winning condition holds. This does not follow from the fact that the expected value of an individual bet is positive. For your betting strategy, for example the event $$\omega = -1, -1, ...$$ will trivially won't let you win. This is also true for the event $$\omega = -1, -1, -1, 1, -1, -1, -1, 1,...$$ etc. So there are non-winning events $\omega_i$ for your betting strategy: Has the set of all such events probability zero?

crawl-data/CC-MAIN-2019-39/segments/1568514575484.57/warc/CC-MAIN-20190922094320-20190922120320-00250.warc.gz

# Making geometric figures (a nifty trick with text orientation and cell merging) Last week I promised that I’d discuss how to use text orientation in spreadsheets to make regular geometric figures. To start off with, a regular geometric figure (also known as a regular polygon) is any figure where all the sides are the same length, and all the angles are the same size. A square is a regular polygon, but a rectangle isn’t because the sides aren’t all the same. Now, I don’t know about you, but the only regular polygon I could have a prayer of actually drawing by hand would be a square, and that’s only because I’d use a ruler and the corner of the sheet of paper to cheat. However, I remember needing a pentagon when I was trying to draw a soccer ball for a report on Germany, and any time you want to put up a “stop” sign you need an octagon. My mom also used to cut geometric shapes out of felt or construction paper, and we could use them to build pictures. I don’t have any kids myself so I don’t know all the possibilities, but I’m sure there’s a fair number of children’s librarians, teachers, and even parents who’ll be able to attest that these shapes come up more often than you’d think when dealing with kids. These days you can always try to look up a template online, but one advantage of being able to do it in a spreadsheet is that you can control the size, line weight, etc. So, without further ado, here’s how you do it. Step 1: Make a line of dashes. This line should be as long as you want each of the sides to be. You might need to start the line with a single quote (‘) to keep the spreadsheet from thinking you’re trying to write a negative number. If you want to make it bold or change the font size, do that too. Step 2: Move/Copy the line to the appropriate number/location of cells. For a triangle you’ll want cell A1, B1, and the merged cell A2&B2. Any time you have an odd number of sides, you’ll need at least one merged cell since there’s going to be a vertex opposite a side. For a square the simplest pattern would be A2, B1, C2, and D1. There are, however, other ways to do it. For a pentagon the pattern is pretty complex, but it makes sense if you think about it carefully. The bottom three lines are going outward, and the top two are going inward. Therefore you’re at least going to need 3 cells on the bottom, and 2 on the top. However, in order for the top vertex to be directly over the bottom line, the bottom line has to be in a merged cell, meaning that there must be 4 cells in the bottom. So the lines go in merged a1&b1, merged c1&d1, a2, merged b2&c2, and d2. For a hexagon you get to do the simplest pattern yet: a1,b1,c1,a2,b2,c2. For an octagon you also get a relatively simple pattern: a1,b1,c1,a2,c2,a3,b3,c3. Step 3: Calculate (or look up) the interior angles. There’s a simple formula for figuring out the interior angles of a regular polygon. It’s 180-(360/n) where n is the number of sides. If you’re curious, it’s because the sum of the supplementary angles to the interior angles of a regular polygon always add up to 360 degrees. This means the angles for a triangle are all 60 degrees, squares are 90 degrees, pentagons are 108, hexagons are 120, and octagons are 135. Step 4: Calculate how angles convert to the direction in a cell. Bottoms for all polygons are exactly the same: 0 degrees. You don’t have to do anything for them. Tops for even-sided polygons also stay at 0 degrees. The sides on a square, and the center-sides on an octagon are both 90 degrees. This will be true of any polygon with a number of sides that’s divisible by 4, just as any polygon with an even number of sides has a top that’s at angle 0. The top left side for odd-sided polygons is It’s 90-(i)/2, where i is the interior angle. So for a triangle it’s 90-60/2 = 90-30 = 60, and for a pentagon it’s 90-108/2 = 90-54 = 36. The top right side for odd-sided polygons can be gotten from that. If you’re in a program that goes all 360 degrees, it’s 360 minus the top left angle (the one you just calculated). If you’re in a program that only goes -90 to 90 degrees it’s even easier: its the negative of the top left angle. The bottom right side of any polygon is just 180 minus the interior angle. (This even works for triangles and squares, though the top pair in triangles have already been addressed, as have the sides of a square.) The bottom left side can be calculated from that: Similar to the top right pair, if you’re in a program that goes all 360 degrees, it’s 360 minus the bottom right angle (the one you just calculated). If you’re in a program that only goes -90 to 90 degrees it’s even easier: its the negative of the bottom right angle. Upper right in hexagons and octagons is the same as the bottom left. Upper left in hexagons and octagons is the same as the bottom right. Step 5: Change the directions of the lines to what you calculated for each. The lines might cross each other, or might not meet, and that’s okay. Step 6: Change the alignment of the lines, if necessary. You want to have the best chance of meeting possible. Therefore if you have two sides that are touching the bottom edge of their cells, then the line in between them needs to be bottom aligned, and probably should be center-aligned as well. It depends on your exact setup, but with the patterns I’ve described here the lines on the left should be right-aligned, the lines on the right should be left-aligned, the lines on the sides should be middle-aligned (vertically), the line on the bottom (and top, for even-sides) should be center-aligned (horizontally), and it should be obvious what the vertical alignments of the top & bottom cells should be once you look at them. Step 7: Adjust the column widths so that the ends of the lines meet each other (or as closes as possible). If necessary, also adjust the row height. Congratulations! You now have a regular geometric figure.

crawl-data/CC-MAIN-2019-09/segments/1550247481612.36/warc/CC-MAIN-20190217031053-20190217053053-00114.warc.gz

A Colonial Christmas in Williamsburg Find more lesson plans, resources, discussion, and more at our new Teacher Community website. Membership is free. Join today! Feasting and celebration were a big part of the Christmas season in colonial Virginia. Then, as now, families and friends gathered to celebrate the holiday with the best their tables could offer. In eighteenth-century Virginia, the holiday season began on December 24 and ran through Twelfth Night on January 6. For centuries, Twelfth Night was the highlight of the holiday season. Although this celebration was not deeply rooted in the American colonies, in the eighteenth century it was celebrated in Virginia, Maryland, Delaware and Pennsylvania. Three to four lesson periods of 60 minutes each - Graphic Organizer: Christmas Then & Now - Receipts and Other Suggestions for a Twelfth Night Celebration (note: receipts is an eighteenth-century term for recipies) - Quill pens (optional) As a result of this lesson, the student will be able to: - plan a colonial Christmas event - demonstrate colonial Christmas customs Setting the Stage Discuss the following with students. There are no eighteenth-century sources which highlight the importance of Christmas to children in particular. In a diary entry of Philip Vickers Fithian dated December 18, 1773, he tells about "the Balls, the Fox-hunts, the fine entertainments." None was meant for kids, and the youngsters were not invited to attend. The emphasis on Christmas as a magical time for children came about in the nineteenth century. So what did the children of Williamsburg do to celebrate Christmas in the eighteenth century? If they were old enough, they might attend church, stick some holly on the window panes, help prepare a great dinner, go to a party, and perhaps visit friends. Give each student a copy of the Graphic Organizer: Christmas Then & Now. Have students work in three (or six if you prefer) collaborative groups. Each group will self-appoint a group leader, a recorder and a reporter. Group I will investigate colonial decorations; Group II will explore eighteenth-century foods; and Group III will look into eighteenth-century singing. After students have read the primary sources on their topic, have the group leader facilitate a discussion on similarities and differences between today and the eighteenth century. Ask the group recorder to fill in the team's responses on the Graphic Organizer: Christmas Then & Now. Ask the group reporter to report to the class the group's findings. Next, have students plan a Twelfth Day of Christmas party by working in three collaborative groups. Group I can decorate the classroom with sprigs of holly in the windows. Real holly or student-created/drawn and colored holly can be used. If real holly is used, have students be authentic by using wax to adhere the holly to the window panes. Group II can plan the food with simple fare, e.g. wassail and gingercakes. The receipts provided in Receipts and Other Suggestions for a Twelfth Night Celebration are two good recipes to use. Other authentic receipts can be found in Recipes from the Raleigh Tavern Bake Shop. Group III can write (using quill pens, if desired) invitations to one another, other classes and/or parents. The model invitation provided in "Receipts and Other Suggestions for a Twelfth Night Celebration" will give students an idea on how to write their invitation. After all preparations are complete, enjoy an eighteenth-century holiday with your students. Have students begin with singing, break for a short repast, and continue with more singing. The date is December 25, 1778. Ask students to assume the role of an American solider in the war for Independence. They are to write a letter home and describe how they spent Christmas as a soldier. (Hint: have them write about things they were not able to do.) Writing as an eighteenth-century child, each student creates a journal describing how they assisted their parents in preparing for a party in celebration of the twelfth day of Christmas. Each journal entry should describe making plans, inviting friends, decorating, and participating in this Twelfth Night party. Encourage students to include illustrations in their journals. Teacher and/or parents can prepare journal pages by soaking onion skin typing paper in coffee and allowing to dry. Then, using a sewing machine, stitch three sheets together by sewing papers in half widthwise. [Optional: Students may use quill pens to make journal entries.] This lesson plan was developed by Carol Mason of Tiffany Elementary School, Chula Vista, California; Glenna Raper, Davis Elementary School, Davis, Oklahoma; and the staff of Colonial Williamsburg's Department of School & Group Services. Learn more about Christmas in colonial Virginia.

<urn:uuid:d593b7e2-1d70-4e46-a953-13a7d8fa239d>

If you are planning to build new roads or expand an existing road, you must know how it will affect the environment. Road building can cause noise, environmental damage, and even harm the wildlife habitats on and around the road. This article will address these topics, as well as the multiscale studies that have been done to uncover the effects of road building on the environment. The environmental impact of road building can be complex. Roads and other structures can have adverse effects on air quality, water quality, and fisheries. In some cases, these effects may be dissociable from the surrounding land use. A number of studies have explored how roads have affected the environment, and the ways in which they have influenced the local acoustic environment. Generally, there are three main areas of focus: the sound of transportation, the noise of construction, and the effect of changes in topography. Transportation is the most pervasive source of noise. For example, a truck traveling at 65 miles per hour sounds twice as loud as a car going at the same speed. Construction noise is rarely the cause of a crisis, but it can be detrimental to wildlife and the environment in general. It is also common for residences to be affected by highway traffic noise. Construction noise can be minimized through appropriate mitigation measures. There are several steps to take, such as installing noise insulation, traffic control devices, and building noise barriers. Depending on the severity of the problem, the most effective methods of mitigation can include the following: Traffic noise is a major contributor to air pollution. Various sources of noise are responsible for this, including trucks, aircraft, trains, and boats. To avoid this, traffic control devices can be installed to prevent trucks from frequently accelerating. Also, the FHWA’s Roadway Construction Noise Model can predict construction noise levels. Aside from the noise, there are other effects of roads. For instance, highway noise affects the physiology and reproduction of animals. Animals may need to be protected from road noise in order to survive. Moreover, animals may have to use more effort to hear. Other impacts of roads can be found at a much smaller scale. In some cases, road-associated changes are quite modest, such as changes in the nutrient cycling of plants or altering local climatic conditions. However, other effects are more significant. One of the more interesting aspects of road building is the interaction it has with the environment. Several factors play a role in determining the road’s effects on the surrounding area, such as the location, type of traffic, and traffic volume. In an urban landscape, wetlands are the foundation of green infrastructure. They provide a diversity of plant life, recreational opportunities, and water supply. However, they are also impacted by human activities and loss of habitat. The effects of road construction on wetlands vary with the spatial scale of the area affected. These effects can range from within a single segment of the roadway to the larger scale of the entire network. For example, a paved road increases runoff and the volume of water reaching the shoreline. This is likely to affect aquatic habitats, riparian systems, and other hydrological processes. Roads have been shown to adversely affect both animal and plant species. They can fragment aquatic ecosystems, increase dispersion, and influence land-use patterns. Moreover, they can contribute to the spread of exotic organisms and reduce soil fertility. Most studies on the impact of roads on wetlands have focused on a single-segment scale. A few have explored a variety of other effects, including chemical pollutants that are long-lived and can reach far into the environment. While most studies focus on single-segment effects, there is little research on the effects of roads at large spatial scales. As a result, there is no general consensus about the best scale for conducting an ecological study on roads. Ultimately, the best scale of assessment depends on the ecological condition that is of interest. Some studies have examined how road building impacts the environment by measuring the number of plants and animals that are affected by it. Others have investigated how roads influence abiotic factors in the environment. While some have been conducted at the project level, most have been at the local level. Although these assessments are important, they have been limited by lack of funding and other factors. Despite these limitations, the current body of research makes important contributions to the knowledge base. The synthesis of information and the analysis of the available data can help determine the magnitude of the effects of roads at a larger scale. It is clear that the potential for wetlands to be used as green infrastructure is not being realized as rapidly as it could. This is due in part to the low level of coordination among researchers and government agencies. There is a need for more research on this topic. Wildlife habitats along roads and bridges One of the most important things that humans can do to help wildlife is to build structures to accommodate the animals. These structures are called wildlife passages or wildlife bridges. They can be a traditional structure or an underpass. Many species need a corridor to reach their home. However, they are often cut off by roads. When roads are built, they destroy habitats and divide the landscape. This impacts the local populations. In addition, many species have special habitat requirements. For instance, pronghorns are known to migrate 125 miles from Grand Teton National Park to Pinedale, Wyoming, every year. A series of overpasses in the Northern Great Plains allows them to cross the highways in a safe manner. Other types of wildlife crossings include modified culverts, wildlife overpasses, and wildlife underpasses. The design for these structures depends on the type of animal using the structures. Some animals are willing to use them, while others aren’t. Wildlife crossings are also a way to connect fragmented wildlife habitat. In the United States, more than 1 million automobile accidents involve wildlife annually. Creating a safe route for animals is important for both the safety of the animals and the safety of the motorists. While wildlife crossings can help mitigate the ecological impact of highways, they can be costly. However, they can pay for themselves in a matter of years. If the highways are well designed and the animals use them safely, they can pay for themselves in 15 years. Although wildlife bridges are not a viable solution for the majority of road-disturbed areas, they can serve as a symbol of the connection between human society and nature. In addition, they can reduce the amount of deadly collisions between drivers and wildlife. Many of these crossings are funded by federal agencies. They are designed to mimic the landscape and are covered with native vegetation. Typically, they are made of steel or cement. These crossings make a great deal of sense for the long-term welfare of both humans and animals. Moreover, they are becoming increasingly popular throughout the country. Multiscale studies uncover the effects of roads on the environment Roads affect both biotic and abiotic components of ecological systems. This can result in changes in biodiversity, habitat and water quality. While many of these effects occur at a local scale, roads can also influence larger spatial and temporal scales. Environmental effects of roads can vary widely depending on road location, topography, density of road network, traffic volume and other factors. For example, less habitat fragmentation occurs in roads with high traffic volumes, and roads can restrict wildlife migration or help connect isolated populations. Studies at various scales have been performed to understand the ecological effects of roads. However, most research has focused on small sampling periods, and this may not adequately sample the variability of ecological systems. Research should focus on the ecological effects of roads at larger spatial and temporal scales. In addition, studies should address the complex nature of roads within ecologically defined areas. The ecological effects of roads are often referred to as “cross-scale effects”. These effects accumulate over time, manifesting as increases in populations, species and ecosystem goods. The most common ecological effects of roads are changes in species composition, water quality and hydrogeomorphic processes. Roads can also contribute to the spread of exotic organisms. Moreover, changes in nutrient cycling can reduce flood mitigation capabilities. Stream habitats are also affected by pollutants in surface runoff. In addition, road operation and maintenance have effects on the environment. Roads change hydrological pathways, and they alter soil and groundwater recharge. Roads also disrupt riparian systems and aquatic habitats. Studies of road effects are mostly conducted at the project level, and most of them have been conducted in the United States. There has been little or no collaborative research among multiple government agencies. Most studies on ecological effects of roads are not included in scientific abstracting services or searchable databases. The literature review provides an overview of current knowledge about the environmental effects of roads. It discusses trends and gaps in the literature. It provides an annotated bibliography of published studies on road effects. The review also addresses the complexity of the effects of roads on the environment. There is still much research to be done. It’s undeniable that roads are necessary for human civilization to thrive, hence road building and Screw Pile Pros Installation should be done by professionals to avoid any unnecessary harm to the environment.

<urn:uuid:5ab0071d-f00f-4c45-84df-508f3ac1773c>

Ultraviolet (UV) light is electromagnetic radiation with a wavelength shorter than that of visible light, but longer than X-rays, in the range 10 nm to 400 nm, and energies from 3 eV to 124 eV. It is named because the spectrum consists of electromagnetic waves with frequencies higher than those that humans identify as the color violet. These frequencies are invisible to humans, but visible to a number of insects and birds. They are also indirectly visible, by causing fluorescent materials to glow with visible light. UV light is found in sunlight and is emitted by electric arcs and specialized lights such as black lights. It can cause chemical reactions, and causes many substances to glow or fluoresce. Most ultraviolet is classified as non-ionizing radiation. The higher energies of the ultraviolet spectrum from wavelengths about 10 nm to 120 nm ('extreme' ultraviolet) are ionizing, but this type of ultraviolet in sunlight is blocked by normal dioxygen in air, and does not reach the ground. However, the entire spectrum of ultraviolet radiation has some of the biological features of ionizing radiation, in doing far more damage to many molecules in biological systems than is accounted for by simple heating effects (an example is sunburn). These properties derive from the ultraviolet photon's power to alter chemical bonds in molecules, even without having enough energy to ionize atoms.

<urn:uuid:800f1e79-5685-4195-85bf-9df4d666b422>

<meta http-equiv="refresh" content="1; url=/nojavascript/"> # Area and Perimeter of Rectangles ## Area is base times height, while perimeter is the sum of the sides. % Progress Practice Area and Perimeter of Rectangles Progress % Area and Perimeter of Rectangles On the morning after the groups were decided and the food was packed up, the groups split up into the backcountry. Kelly’s group began hiking up the Lonesome Lake Trail which starts at the base of the mountains in Lafayette Place Campground. They were carrying tons of gear, but were very excited. Their group leaders Scott and Laurel told them that their first stop would be the Lonesome Lake Hut. The AMC has huts stationed all through the White Mountains. Some of them are even at the top of mountains. These huts have beds, some have full kitchens, some have self-service kitchens, but hikers of all different kinds stay at the huts. The Lonesome Lake Hut is located at 2,760 feet and the trail is 1.5 miles. It would be one of the easier hikes they would go on. Scott told the group that they would be helping to repair the floor of the main house. It seems that there was some water damage. The roof has been repaired but the floor needs work. The group hiked the beautiful, relatively easy trail in about an hour and a half. When they got there, they were delighted to see everything. After unpacking in a bunkhouse and getting some lunch, the hikers got to work. The crew leader who works at the hut told the students that the floor was damaged during the winter. The area of the floor is 90,720 square feet and one quarter of the floor would need to be fixed. That is an area of 22,680 square feet and a length of 90 ft wide. If the length is 90 ft wide and the area is 22,680 square feet, what is the width of the area that needs repair? To figure this out, you will need to learn about formulas and working backwards. Pay attention and you will know how to solve this problem at the end of the Concept. ### Guidance Measurement can also be completed with specific figures and formulas. Once we obtain measurements of the length and width of squares and rectangles, we are able to use formulas to find perimeter and area. What is perimeter and area? What does each of them measure? Perimeter is the distance around a figure. Area is the square units inside a figure. Because perimeter is the distance around a figure, we can find the perimeter of a rectangle or square by adding the lengths of all four sides. A rectangle has two lengths that are equal and two widths that are equal, so the perimeter of a rectangle can be described by the formula $P = 2l + 2w$ . Remember that a square has four equal sides. Therefore, it is possible to find the perimeter of a square by multiplying one of the sides times 4. The formula for finding the perimeter of a square is $P = 4s$ where $s$ is the measurement of one side. Area is the square units inside a figure; it describes how much space a rectangle or square takes up. To find the area of rectangle, we multiply the length times the width. $A = lw$ . A square has four equal sides, so when finding the area of a square, all we need to do is multiply one of the sides by itself. $A = s^2$ . Sure. Here they are once again. $&\text{Rectangle} && P = 2l + 2w\\&&& A = lw\\&\text{Square} && P = 4s\\ &&& A = s^2$ Write these formulas down in a notebook before continuing with the Concept. Find the perimeter and area of the following rectangle. First, let’s find the perimeter of the rectangle. To do this, we take our formula and substitute the measure for length in for $l$ and the measure for width in for $w$ . $P &= 2l+2w\\P &= 2(12)+ 2(7)\\P &= 24+14\\P &= 38 \ inches$ Now, we can find the area of the rectangle. To do this, we take the formula for area and substitute the given values in for length and width. $A &= lw\\ A &= (12)(7)\\A &= 84 \ sq.in.$ Use the formulas and find the perimeter and area of the following square. Here you can see that we have only been given the length on one side. That is okay though because this is a square. A square has four congruent or equal sides. Therefore, all we need is one side length to work with. First, let’s find the perimeter of the square. $P &= 4s\\ P &= 4(14)\\ P &= 56 \ feet$ Next, we can use the formula for area to find the area of the square. $P &= s^2\\ P &= 14^2=14(14)\\ P &= 196 \ square \ feet \ or \ ft^2$ Now try a few on your own. #### Example A Find the perimeter and area of a rectangle with a width of 10 inches and length of 12 inches. Solution: 44 inches #### Example B The perimeter of a square with a side length of 4.5 inches. Solution: 18 inches #### Example C The perimeter of a rectangle with a length of 15 feet and a width of 12 feet. Solution: 54 feet Now back to the shelter at Lonesome Lake. Here is the original problem once again. On the morning after the groups were decided and the food was packed up, the groups split up into the backcountry. Kelly’s group began hiking up the Lonesome Lake Trail which starts at the base of the mountains in Lafayette Place Campground. They were carrying tons of gear, but were very excited. Their group leaders Scott and Laurel told them that their first stop would be the Lonesome Lake Hut. The AMC has huts stationed all through the White Mountains. Some of them are even at the top of mountains. These huts have beds, some have full kitchens, some have self-service kitchens, but hikers of all different kinds stay at the huts. The Lonesome Lake Hut is located at 2,760 feet and the trail is 1.5 miles. It would be one of the easier hikes they would go on. Scott told the group that they would be helping to repair the floor of the main house. It seems that there was some water damage. The roof has been repaired but the floor needs work. The group hiked the beautiful, relatively easy trail in about an hour and a half. When they got there, they were delighted to see everything. After unpacking in a bunkhouse and getting some lunch, the hikers got to work. The crew leader who works at the hut told the students that the floor was damaged during the winter. The area of the floor is 90,720 square feet and one quarter of the floor would need to be fixed. That is an area of 22,680 square feet and a length of 90 ft wide. If the length is 90 ft wide and the area is 22,680 square feet, what is the width of the area that needs repair? First, we let’s write an equation to solve this problem. We can assume that the hut is a rectangle in shape because we were given a length and asked to find the width. $A &= lw\\22680 &= 90w$ We need to find a number that when multiplied by 90 is equal to 22,680. To do this, it makes sense to divide. Division is the opposite of multiplication. Since mental math won’t work for this one, division is the next likely option. $22680 \div 90 = 63 \ ft$ The width of the area of damaged floor is 63 feet wide. The students worked hard to repair the floor and it was a great two days working at Lonesome Lake Hut! Yet when their work was done they were excited to think about hiking to their next destination! ### Guided Practice Here is one for you to try on your own. Marcy has purchased a new rug for her bedroom. The rug measures 6 ft. x 9 ft. Given these measurements, what is the perimeter of the rug? What is the area? Because you were given the measurements 6' x 9', you can deduce that this rug is rectangle. Therefore we can find the perimeter of the rug by doubling the length and width and then add them together. $P = 2(6) + 2(9)$ $P = 30$ feet The area of the rug can be found by using the following formula. $A = lw$ $A = (6)(9)$ $A = 54$ sq. feet. ### Explore More Directions: Find the area and perimeter of each rectangle by using the given dimensions. 1. Length = 10 in, width = 5 in 2. Length = 12 ft, width = 8 feet 3. Length = 11 ft, width = 5 feet 4. Length = 17 miles, width = 18 miles 5. Length = 22 ft, width = 20 feet 6. Length = 8 cm, width = 6 cm 7. Length = 20 cm, width = 17 cm 8. Length = 3 feet, width = 2 feet 9. Length = 15 yards, width = 11 yards 10. Length = 10 yards, width = 6 yards Directions: Find the area and perimeter of each square using the given dimensions. 11. $s = 6 \ ft$ 12. $s = 8 \ ft$ 13. $s = 9 \ in$ 14. $s = 4 \ in$ 15. $s = 12 \ in$ 16. $s = 7 \ ft$ 17. $s = 5 \ cm$ 18. $s = 3 \ m$ 19. $s = 10 \ m$ 20. $s = 11 \ yards$ ### Vocabulary Language: English Area Area Area is the space within the perimeter of a two-dimensional figure. Perimeter Perimeter Perimeter is the distance around a two-dimensional figure. Area of a Rectangle Area of a Rectangle To find the area 'A' of a rectangle, calculate A = bh, where b is the base (width) and h is the height (length). Perimeter of a Rectangle Perimeter of a Rectangle The perimeter 'P' of a rectangle is equal to twice the base added to twice the height: P = 2b + 2h.

crawl-data/CC-MAIN-2015-22/segments/1432207928864.73/warc/CC-MAIN-20150521113208-00071-ip-10-180-206-219.ec2.internal.warc.gz

Make a Bar Graph: Bat Sightings Let's graph! Mathematicians review how to make a bar graph using given data. They use data about the number of bats a child saw, and then they use the data to answer three questions. They solve a word problem based on the same graph and explain their answer.

<urn:uuid:5db3be32-cae6-4409-af86-c9635ca8780b>

Westward Expansion and Sectionalism (1840-1861) Essay Westward Expansion and Sectionalism (1840-1861) At the end of the Mexican War during Polk’s term as president, many new lands west of Texas were yielded to the United States, and the debate over the westward expansion of slavery was rekindled. Southern politicians and slave owners demanded that slavery be allowed in the West because they feared that a closed door would spell doom for their economy and way of life. Whig Northerners, however, believed that slavery should be banned from the new territories. Pennsylvanian congressman David Wilmot proposed such a ban in 1846, even before the conclusion of the war. Southerners were outraged over this Wilmot Proviso and blocked it before it could reach the Senate. When this act was denied it essentially caused America to become a country of two halves. Sadly, this division caused Americans to provoke wickedness against one another: the North vs. South, Slavery vs. Freedom, and Brother vs. Brother. The Wilmot Proviso justified Southerners’ fears that the North had designs against slavery. They worried that if politicians in the North prevented slavery from expanding westward, then it was only a matter of time before they began attacking it in the South as well. As a result, Southerners in both parties flatly rejected the proviso. Such biased support was unprecedented and demonstrated just how serious the South really felt about the issue. The large land concessions made to the U.S. in the 1848 Treaty of Guadalupe Hidalgo only exacerbated tensions between the North and the South. Debates in Congress grew so heated that even fist fights broke out between Northern and Southern politicians on the floor of the House of Representatives. In fact, sectional division became so evident that many historians label the Mexican-American War and the Wilmot Proviso the first battles that ignited the Civil War. Even though the Wilmot Proviso had failed, the expansion of slavery remained the most demanding issue in the world of politics at the time. The Democrats, meanwhile, nominated Lewis Cass. Also hoping to sidestep the issue of slavery, Cass proposed allowing the citizens of each western territory to decide for themselves whether or not to be free or slave. Cass hoped that a platform based on such popular sovereignty would win him votes in both the North and South. The election of 1848 also marked the birth of the Free-Soil Party, a hodgepodge collection of Northern abolitionists, former Liberty Party voters, and disgruntled Democrats and Whigs. The Free-Soilers nominated former president Martin Van Buren, who hoped to split the Democrats. He succeeded and diverted enough votes from Cass to throw the election in Taylor’s favor. Although Taylor’s silence on the issue quieted the debate for about a year, the issue was The Slavery Debate revived when California and Utah applied for statehood. California’s population had boomed after the 1849 gold rush had attracted thousands of prospectors, while barren Utah had blossomed due to the ingenuity of several thousand Mormons. The question arose whether these states should be admitted as Free states or slave states. The future of slavery in the United States was in the hands of Washington D.C. A great debate ensued in Congress over the future of these three regions as Southerners attempted to defend their economic system while Northerners decried the evils of slavery. In Congress, the dying John C. Calhoun argued that the South still had every right to nullify unconstitutional laws and, if necessary, to secede from the Union it created. Daniel Webster and Henry Clay, on the other hand, championed the Union and compromise. Webster in particular pointed out that discussion over the expansion of slavery in the West was debatable because western lands were unsuitable for growing cotton. In the end, the North and South agreed to compromise. Although Clay was instrumental in getting both sides to agree, he and Calhoun were too elderly and infirm to negotiate concessions and draft the necessary legislation. This task fell to a younger generation of politicians, especially the “Little Giant” Stephen Douglas, so named for his short stature and big mouth. A Democratic senator from Illinois, Douglas was responsible for pushing the finished piece of legislature through Congress. The Compromise of 1850, as it was called, was a bundle of legislation that everyone could agree on. First, congressmen agreed that California would be admitted to the Union as a free state (Utah was not admitted because the Mormons refused to give up the practice of polygamy). The fate of slavery in the other territories, though, would be determined by popular sovereignty. Next, the slave trade (though not slavery itself) was banned in Washington, D.C. Additionally, Texas had to give up some of its land to form the New Mexican territory in exchange for a cancellation of debts owed to the federal government. Finally, Congress agreed to pass a newer and tougher Fugitive Slave Act to enforce the return of escaped slaves to the South. Though both sides agreed to it, the Compromise of 1850 clearly favored the North over the South. California’s admission as a free state not only set a precedent in the West against the expansion of slavery, but also ended the sectional balance in the Senate, with sixteen free states to fifteen slave states. Ever since the Missouri Compromise, this balance had always been considered essential to prevent the North from banning slavery. The South also conceded to end the slave trade in Washington, D.C., in exchange for debt relief for Texans and a tougher Fugitive Slave Law. Southerners were willing to make so many concessions because, like Northerners, they truly believed the Compromise of 1850 would end the debate over slavery. As it turned out, of course, they were wrong. Ironically, the Fugitive Slave Act only fueled the abolitionist flame rather than put it out. Even though many white Americans in the North felt little love for African-Americans, they detested the idea of sending escaped slaves back to the South. In fact, armed mobs in the North freed captured slaves on several occasions, especially in New England, and violence against slave catchers increased despite the federal government’s protests. The Fugitive Slave Act thus allowed the abolitionists to transform their movement from a radical one to one that most Americans supported. Even though few slaves actually managed to escape to the North, the fact that Northern abolitionists encouraged slaves to run away infuriated Southern plantation owners. One network, the Underground Railroad, did successfully ferry as many as several thousand fugitive slaves into the North and Canada between 1840 and 1860. “Conductor” Harriet Tubman, an escaped slave from Maryland, personally delivered several hundred slaves to freedom. Another major boost for the abolitionist cause came via Harriet Beecher Stowe’s 1852 novel Uncle Tom’s Cabin, a story about slavery in the South. Hundreds of thousands of copies were sold, awakening Northerners to the plight of enslaved blacks. The book affected the North so much that when Abraham Lincoln met Stowe in 1863, he commented, “So you’re the little woman who wrote the book that made this Great War!” President Lincoln was correct that this war would be indeed a Great War, a war that would push State against State, the North against the South, and worst of all, Brother against Brother. The Civil War would be a struggle for both the North and the South. It would be six grueling years before peace returns to the country and the American states would once again be a united nation. University/College: University of Arkansas System Type of paper: Thesis/Dissertation Chapter Date: 6 November 2016 Let us write you a custom essay sample on Westward Expansion and Sectionalism (1840-1861) for only $16.38 $13.9/page

<urn:uuid:461c07d6-2271-49bf-8959-230603c7def5>

Rain brings happiness to everyone on earth. Clouds that move across the sky contain huge amounts of tiny water droplets and ice crystals. Not all clouds bring rain. Only under ideal conditions clouds condense into rain. Rain water is very good for plants as it contains lot of nutrients and is free of salts and other harmful elements. Though rain absorbs gases from atmosphere, it is pure till it reaches the ground. Rainwater is soft water and hence it doesn’t form scales on surfaces or utensils. It also lathers well with soaps and detergents. Once rain water reaches the ground, it gets contaminated with chemicals and other water pollutants that are present on the ground. Rain with more than normal amounts of chemicals constituents is called acid rain. Acid rains occur only in the most industrialized regions. It is mainly caused by the presence of harmful gases released into the environment by factories, power plants, automobiles etc. The two major pollutants causing acid deposition are sulfur dioxide (SO2) and nitric oxide (NOx). These chemicals become acids when they enter the air and react with water vapor. When it rains, it brings sulfuric acid (H2SO4) and nitric acid (HNO3) to earth either as wet or dry precipitations. A wet precipitation leads to acid rains and a dry one leads to particulates of acids. While normal rain has a pH in the order of 5-6 on the scale, acidic rains tend to a pH of 3-4. Acid rain causes acidification of lakes and streams and contributes to the damage of trees at high elevations. Rain water harvesting is done in many regions across the globe in order to save the precious resource. There are many methods in which rain water can be harvested. Depending on the type of house and available structures like well, bore wells etc appropriate water harvesting methods are followed.

<urn:uuid:e9cb7634-9c48-4b93-8747-f8ecff2925f3>

# How do you integrate sin(x)*(e)^(2 x) dx? ##### 1 Answer Jul 1, 2016 Use integration by parts (IBP). Solution: $\int {e}^{2 x} \sin \left(x\right) \mathrm{dx} = \frac{2}{5} {e}^{2 x} \sin \left(x\right) - \frac{1}{5} {e}^{2 x} \cos \left(x\right) + C$ #### Explanation: To obtain the formula for IBP, we start with the product rule: $\left(u \cdot v\right) ' = u ' \cdot v + u \cdot v '$ where u and v are functions of x in this case Rearranging: $u \cdot v ' = \left(u \cdot v\right) ' - u ' \cdot v$ Integrating: $\int u \cdot v ' \mathrm{dx} = u \cdot v - \int u ' \cdot v \mathrm{dx}$ For this integral, we are going to set $u \left(x\right) = \sin \left(x\right)$ and $v ' \left(x\right) = {e}^{2 x}$ $u ' \left(x\right) = \cos \left(x\right)$ and $v \left(x\right) = \frac{1}{2} {e}^{2 x}$ So, $\int {e}^{2 x} \sin x \mathrm{dx} = \frac{1}{2} {e}^{2 x} \sin \left(x\right) - \frac{1}{2} \cdot \int \cos \left(x\right) {e}^{2 x} \mathrm{dx}$ Now do IBP again on the second term: $u \left(x\right) = \cos \left(x\right)$ and $v ' \left(x\right) = {e}^{2 x}$ $u ' \left(x\right) = - \sin \left(x\right)$ and $v \left(x\right) = \frac{1}{2} {e}^{2 x}$ $\int {e}^{2 x} \sin \left(x\right) \mathrm{dx} = \frac{1}{2} {e}^{2 x} \sin \left(x\right) - \frac{1}{4} {e}^{2 x} \cos \left(x\right) - \frac{1}{4} \cdot \int {e}^{2 x} \sin \left(x\right) \mathrm{dx}$ As you can see, the integral on the LHS is the same as the integral on the RHS so we collect like terms and obtain: $\frac{5}{4} \cdot \int {e}^{2 x} \sin \left(x\right) \mathrm{dx} = \frac{1}{2} {e}^{2 x} \sin \left(x\right) - \frac{1}{4} {e}^{2 x} \cos \left(x\right)$ $\int {e}^{2 x} \sin \left(x\right) \mathrm{dx} = \frac{2}{5} {e}^{2 x} \sin \left(x\right) - \frac{1}{5} {e}^{2 x} \cos \left(x\right) + C$

crawl-data/CC-MAIN-2022-27/segments/1656103640328.37/warc/CC-MAIN-20220629150145-20220629180145-00050.warc.gz

Fossils are the preserved remains or traces of animals plants and other organisms from the remote past. There are different type of fossils: TRACE FOSSILS: are the remains of trackways, burrows, bioerosion, eggs and eggshells, nests, droppings and other types of impressions. RESSIN FOSSILS: is a natural polymer found in many types of strata throughout the world, even the artic. PSEUDOFOSSILS: are visual patterns in rocks that are produced by naturally occurring geologic processes rather than biologic processes. LIVING FOSSILS: is an informal term used for any living species that is apparently identical or closely resembles a species previously known only from fossils that is, it is as if the ancient fossil had “come to life.”

<urn:uuid:fed1efe4-2353-439a-af43-fe24dd621c1f>

Harvard Study Finds Wind Turbines Will Cause More Warming In Minnesota Than Emissions Reductions Would Avert A 2018 study conducted by scientists from Harvard, published in the academic journal Joule, found that wind turbines cause significant local increases in surface temperatures in the areas where they are located. Wind turbines cause local temperature increases at the surface of the earth by causing air to mix throughout portions of the atmosphere, and Minnesota would be one of the states impacted most by this phenomenon. Warming Wind Turbines According to the study, wind turbines measuring between 100 and 150 meters (328ft to 498 ft) operating at night can pull down warmer air from as far as 1,640 feet in the air down to the surface, warming the surface of the earth, where it would impact the people, plants, and animals living near the turbines. The study looks at what would happen if the United States tried to obtain all of its energy from wind turbines. It found the mixing of warmer air and cooler air results in a temperature increase of 0.54 degrees Celsius (0.97 degrees F) in the areas where the wind turbines would be located, as you can see in the figure below from the study. The amount of warming experienced in some regions would be even greater, as Southwestern Minnesota could see a temperature increase of 0.6-0.8 degrees C due to wind turbines, while Northeastern Minnesota would see an increase of 0.3-0.5 degrees C. Most of the warming would occur at night, according to the study: “The wind farm region experiences warmer average temperatures (Figure 1A), with about twice the warming effect at night compared with during the day (Figures 1B and 1C). Warming was generally stronger nearer to the center of the wind farm region.” All Energy Sources Have Impacts This study is interesting because it acknowledges that all energy sources, whether they be coal, natural gas, wind or solar have environmental impacts. It seems to be one of the few studies that attempts to evaluate the costs of wind turbines, along with their supposed benefits, and use this cost/benefit to figure out which sources of energy have the fewest environmental impacts. The authors claim this will be important when discussing which carbon-free sources of electricity will be used in the future, but I believe it informs our current energy decisions,. According to the study: “The climatic impacts differ in (at least) two important dimensions. First, the direct climatic impact of wind power is immediate but would disappear if the turbines were removed, while the climatic benefits of reducing emissions grows with the cumulative reduction in emissions and persists for millennia. Second, the direct climatic impacts of wind power are predominantly local to the wind farm region, while the benefits of reduced emissions are global.” In other words, the warming impact of wind turbines is immediate, and highly localized in the areas that are the “hosts” to the installations. The supposed benefits of reducing carbon dioxide emissions are global, not local. This means places like Minnesota will see an increase in temperature from wind turbines that exceeds amount of potential future global warming that would be averted from completely reducing Minnesota’s greenhouse gas emissions to zero. Minnesota Emissions Reductions According to the Minnesota Pollution Control Agency, Minnesota emitted about 150 million metric tons of greenhouse gases in 2016. Using the same logic used by the Obama Administration to craft the Clean Power Plan, if we completely eliminated all of these emissions to zero, it would avert only 0.004 degrees C by 2100, which is an amount far too small to measure! In fact, the amount of global warming averted (0.004 degrees C) would be 138 times smaller than the warming Minnesota would incur from building out wind turbines to power all of our electricity use (0.54 degrees C), as you can see in Figure 5(d) from the study below.The orange dotted line shows surface temperature increases in the areas with wind turbines, and the orange solid line shows the temperature impact of wind turbines on the entire continental United States. The blue and grey shaded areas show the differences in surface temperatures in the United States from reducing our national emissions. As you can see, surface temperatures in the United States increase more due to wind turbines mixing air in the atmosphere than would be offset by reducing emissions. This is especially true in areas like Minnesota, where the wind turbines would be operating. The only time that reduced emissions might impact surface temperatures more than the wind turbines, themselves, is if the entire world reduces their carbon dioxide emissions, but if you believe China will actually reduce their emissions, I’ve got a bridge to sell you. In light of this study, it makes zero sense to build wind in Minnesota if our Governor truly wants to “make sure there is still ice on that lake in January,” because surface temperatures will increase much more from the wind turbines than they would fall by reducing emissions. A note to the Governor, increasing surface temperatures would reduce the amount of ice on that lake in January. Center of the American Experiment has been one of the leading organizations advocating for Minnesota to repeal its antiquated ban on new nuclear power plants. We have also been some of the strongest supporters of allowing hydroelectric power that we already purchase from Canada to count toward our renewable energy mandates. If Governor Walz and liberal legislators worry about the impact of global warming on Minnesota, then they need to own up to the fact that the surface temperature impacts of wind turbines mixing air in the atmosphere will far outweigh the amount of warming that would be averted from reducing emissions and seek to legalize new nuclear, large hydro, and promote carbon capture and sequestration technologies that provide reliable electricity without carbon dioxide emissions. In the spirit of full fairness, it should be noted that the findings of this study are based on General Circulation Models (GMS), which overestimate the amount of global warming that is observed with weather balloons and satellites by a factor of two, so the results of this study may well be as legitimate as Governor Walz’s COVID-19 models. However, it should be noted that all of the policies renewable energy specialist interest groups try to pass in Minnesota to avert climate change are also based on GCM’s, so it is highly inconsistent for them to trumpet GCM’s as gospel in one instance (when it promotes something they like), and completely ignore them in another (when their findings conflict with their policy preferences).

<urn:uuid:31d22a58-5f2a-4cb3-8429-a0b11ba1a5ec>

A rare space weather event May 11 marked by a sharp decrease in solar wind helped cause Earth's magnetosphere to balloon to more than 100 times its normal volume, reaching nearly to the moon in the process, according to a new study. Daniel Baker, director of the University of Colorado at Boulder’s Laboratory for Atmospheric and Space Physics, said the density and speed of the particle-toting solar wind creates a dynamic pressure that confines Earth's magnetic field to form its magnetosphere. Under normal conditions, a cross-section of Earth’s magnetosphere resembles a halved apple, although a comet-like tail sweeps back from the side of the planet facing away from the sun. But on May 11, the solar wind density – the number of energetic electrons and protons per cubic centimeter -- was only about 2 percent of normal, and the wind’s speed dropped by more than half. This caused the pressure on the magnetic field to plummet by more than 99 percent, said Baker, lead author on the study. The event was dubbed by some scientists as "The Day the Solar Wind Ran Out of Gas." "This was a very rare occurrence that has been seen only a few times since satellites began taking solar wind measurements 35 years ago," he said. While the magnetosphere generally reaches out about 40,000 miles from Earth toward the sun, it stretched to nearly 235,000 miles on May 11, about the distance from the Earth to the moon. A paper on the subject by Baker, CU graduate student Sean Yarborough and CU Research Associate Xinlin Li was presented at the fall meeting of the American Geophysical Union held Dec. 13 to Dec. 17 in San Francisco. Other co-authors included Niescja Turner and Shrikanth Kanekal of NASA's Goddard Space Flight Center, J.B. Blake of the Aerospace Corp. in Los Angeles and Howard Singer of NOAA’s Space Environment Center in Boulder. NOAA is the National Oceanic and Atmospheric Administration. Baker and his colleagues used data from the SAMPEX, POLAR, GOES, GPS and many other satellites to take data on Earth's Van Allen radiation belts, which are deeply embedded in the magnetosphere. The radiation belts became much more symmetric during the event, with the comet-like tail of radiation apparently disappearing in the process. Although the density of energetic electrons in the solar wind returned to normal on May 12, as did the solar wind speed, the density of very high-energy electrons in the magnetosphere dropped mysteriously once again on May 13 and remained "severely depleted" for an extended period, said Baker. "It appears this episode fundamentally changed the magnetosphere by causing the energetic electrons to all but disappear for about two months," said Baker. "Everything the solar wind was throwing at the magnetosphere should have re-started its engine after May 13, but for some reason it failed." Although scientists once thought Earth’s radiation belts slowly waxed and waned and were not particularly dynamic, they recently have been shown to be powerful particle accelerators for the magnetosphere’s energetic electrons, he said. The solar wind spins off the rotating sun much like a circular lawn sprinkler sprays water, accelerating up to nearly 1 million to 2 million miles per hour by the time it hits Earth's magnetosphere, said Baker. "It’s possible that a large 'vacuum bubble’ formed on the edge of the sun and was blown toward Earth, causing the solar wind to slow and become less dense during the May 11 event," he said. "This illustrated the solar wind, like the radiation belts, are highly dynamic and variable." Although the shape of the magnetosphere and the density of its charged particles returned to normal levels after about two months, the surprising 1999 phenomenon may prove significant to space-weather experts like Baker. "I think these rare, extreme events eventually will help us to understand how the magnetosphere's engine works under normal circumstances," he said. Cite This Page:

<urn:uuid:a881980f-62fa-4e7d-b3b5-014e69d7df75>

# If the value of 2x - $$\frac{1}{2x} = 6$$, then the value of 4x2+ $$\frac{1}{4x^2}$$ is This question was previously asked in UPPSC Civil Service 2016 Official Paper 2 View all UPPCS Papers > 1. 38 2. 39 3. 40 4. 42 Option 1 : 38 ## Detailed Solution Given: If the value of 2x - $$\frac{1}{2x} = 6$$ Formula used: (a - b)2 = a2 + b2 - 2ab Calculation: 2x - $$\frac{1}{2x} = 6$$ (a - b)2 = a2 + b2 - 2ab ⇒ (2x - 1/2x)2 = 4x2 + 1/4x2 - 2(2x)(1/2x) ⇒ 62 = 4x2 + 1/4x2 - 2 ⇒ 36 + 2 = 4x2 + 1/4x2 ⇒ 4x2 + 1/4x2 = 38 ∴ The value of 4x2 + 1/4x2 is 38.

crawl-data/CC-MAIN-2021-39/segments/1631780055632.65/warc/CC-MAIN-20210917090202-20210917120202-00177.warc.gz

What is the GPRS full form? GPRS stands for general packet radio service. It is based on GSM infrastructure. Becomes very popular, because of access to internet over wireless medium. A mobile phone with sim card provides voice and internet services to the user. With this service, internet packets flow from device to the internet service provider. Enables a user to browse websites , make voip calls, connect on VPN etc. Initially telecom network was supporting only voice, then a separate core network was developed for mobile data or packer service over GSM. Full form says that GPRS enables packet based service over radio interface. Which means over mobile device which connects to the mobile network over radio interface. How GPRS works ? Network for packet service: As like any other service over GSM, GPRS also uses standard protocols and procedures. These are define in European Telecommunications Standards Institute(ETSI). Mobile devices having settings for GPRS profile and mobile network have GPRS settings configured in data base. On roaming side a dedicated node called SGSN added for GPRS registration and internet packet flow from device to home network and vice versa. In LTE, the node is replaced with MME. In home network, HLR updated with GPRS functionalities an new node GGSN has been added. For HLR, HSS and for GGSN , PGW are the nodes in LTE network. In 3G architecture SS7 protocol and GTP ( version 0 and 1) are used , while in 4G Diameter protocol and GTP (version 1 and 2 ) are used. GPRS Profile settings: To access internet a mobile subscriber should have GPRS subscription. HLR is the node which holds the subscription. The subscription includes , Quality of service, access point name etc. Upon GPRS registration by the phone, roaming network downloads the subscription profile. GPRS internet speed : The data speed depends on the generation using GPRS network. For 2G its slowest and 4G its fastest. The next network is 5G that will brig more data speed over GPRS.

<urn:uuid:03f7ffb7-7ad7-4c17-b29e-1e097237c693>

What is the Prime Factorization of 126? Learn how to find the prime factorization of 126 with this comprehensive guide. Discover the fundamental concept of prime factors and how to apply them in mathematics. Introduction Prime factorization is a fundamental concept in mathematics that breaks down a composite number into its prime factors. It’s an essential tool in solving various mathematical problems, including finding the greatest common factor, simplifying fractions, and solving equations. In this article, we’ll be discussing the prime factorization of 126. Definition of Prime Factorization Prime factorization is the process of breaking down a composite number into its prime factors. A composite number is a positive integer greater than one that has at least one positive divisor other than one or itself. Prime factors are prime numbers that divide a composite number without leaving a remainder. In other words, prime factors are the building blocks of a composite number. For instance, the prime factorization of 24 is 2 x 2 x 2 x 3. This means that 24 is the product of 2 raised to the power of 3 and 3 raised to the power of 1. By writing a number as a product of its prime factors, it becomes easier to work with and manipulate mathematically. Now that we understand what prime factorization is, let’s dive into how to find the prime factorization of 126. (to be written) How to Find Prime Factorization of 126 To find the prime factorization of 126, we can use the factor tree method. Here’s a step-by-step process to find the prime factorization of 126: 1. Start by dividing the number by the smallest prime number, which is 2. 2. Keep dividing by 2 until it’s no longer divisible by 2. 3. Then, divide the resulting number by the smallest prime factor, which is 3. 4. Keep dividing by 3 until it’s no longer divisible by 3. 5. Continue this process until the remaining number is itself a prime number. Maybe you are interested  What Channel is NBC in Utah? Find Out Here Using this method, we get the prime factorization of 126 as 2 x 3 x 3 x 7. Another way to write the prime factorization of 126 is by using exponents. We can write 126 as 2 x 3² x 7, which means that 126 is the product of 2 raised to the power of 1, 3 raised to the power of 2, and 7 raised to the power of 1. Factors of 126 To find the factors of 126, we need to find all the numbers that divide 126 without leaving a remainder. The factors of 126 are 1, 2, 3, 6, 7, 9, 14, 18, 21, 42, 63, and 126. We can identify the prime factors of 126 by listing its factors and then selecting only the prime factors. The prime factors of 126 are 2, 3, and 7. This means that 126 is the product of 2 raised to the power of 1, 3 raised to the power of 2, and 7 raised to the power of 1. Knowing the prime factorization of 126 and its factors can help us solve various mathematical problems, such as finding the greatest common factor and least common multiple of two or more numbers. Now that we’ve discussed the prime factorization and factors of 126, let’s move on to writing the prime factorization of 126 using exponents. Prime Factorization of 126 To find the prime factorization of 126, we need to first identify the prime factors of 126. We can do this by dividing 126 by the smallest prime number, which is 2. 126/2 = 63 We can see that 2 is not a factor of 63. Instead, 3 is the next smallest prime number. Maybe you are interested  What Size is a 2 Quart Baking Dish? 63/3 = 21 Again, 3 is not a factor of 21. We can divide 21 by 3 again. 21/3 = 7 Now, we have found the prime factors of 126, which are 2, 3, and 7. Therefore, the prime factorization of 126 is 2 x 3 x 3 x 7. We can also write the prime factorization of 126 using exponents. To do this, we write each prime factor as a base with its exponent being the number of times it appears in the factorization. 2 x 3 x 3 x 7 can be written as 2^1 x 3^2 x 7^1. Conclusion In conclusion, prime factorization is a critical concept in mathematics that helps in solving various mathematical problems. In this article, we discussed the prime factorization of 126, which is 2 x 3 x 3 x 7 or 2^1 x 3^2 x 7^1. By breaking down composite numbers into their prime factors, we can simplify mathematical problems and make them more manageable. How Many Pounds is 600 kg? – A Comprehensive Guide Learn how to convert 600 kg into pounds with our comprehensive guide. Discover the factors that impact weight measurement, including gravity, altitude, and temperature. How Many Pounds is 600 kg? – A Comprehensive Guide Learn how to convert 600 kg into pounds with our comprehensive guide. Discover the factors that impact weight measurement, including gravity, altitude, and temperature. What is the Optional DC Cable for the Yaesu FT-70DR? Learn how to use the optional DC cable for the Yaesu FT-70DR to ensure uninterrupted radio communication in the field. Find out what it is and how to use it! How Many Ounces in 1.5 Pounds? Learn how to convert pounds to ounces and vice versa accurately! Discover “how many ounces in 1.5 pounds” and more with this comprehensive guide. What is Rebirth 2k22? Unlocking the Mysteries of Reincarnation Discover the mysteries of rebirth 2k22 – the concept of reincarnation in the year 2022. Explore its significance in different beliefs and how to achieve it. What is a Smoochie Girl? Discover the truth about what a smoochie girl is and the harmful effects of this behavior. Learn how to break free from the cycle and embrace authenticity.

crawl-data/CC-MAIN-2024-33/segments/1722640365107.3/warc/CC-MAIN-20240803091113-20240803121113-00067.warc.gz

The Common Core State Standards Shifts in Mathematics Focus strongly where the Standards focus The Standards call for a greater focus in mathematics. Rather than racing to cover topics in a mile-wide, inch-deep curriculum, the Standards require us to significantly narrow and deepen the way time and energy is spent in the math classroom. We focus deeply on the major work of each grade so that students can gain strong foundations: solid conceptual understanding, a high degree of procedural skill and fluency, and the ability to apply the math they know to solve problems inside and outside the math classroom. Students should spend the large majority of their time on the major work of the grade (). Supporting work () and, where appropriate, additional work () can engage students in the major work of the grade. Emphases are given at the cluster level. Refer to the Common Core State Standards for Mathematics for the specific standards that fall within each cluster. Where to focus by grade level:Download All Grades K-8 As a Single PDF Download All Grades K-HS As Individual PDFs Coherence: think across grades, and link to major topics within grades Thinking across grades: The Standards are designed around coherent progressions from grade to grade. Learning is carefully connected across grades so that students can build new understanding onto foundations built in previous years. Each standard is not a new event, but an extension of previous learning. Linking to major topics: Instead of allowing additional or supporting topics to detract from the focus of the grade, these concepts serve the grade level focus. For example, instead of data displays as an end in themselves, they are an opportunity to do grade-level word problems. Rigor: in major topics pursue: conceptual understanding, procedural skill and fluency, and application with equal intensity. Conceptual understanding: The Standards call for conceptual understanding of key concepts, such as place value and ratios. Students must be able to access concepts from a number of perspectives so that they are able to see math as more than a set of mnemonics or discrete procedures. Procedural skill and fluency: The Standards call for speed and accuracy in calculation. Students are given opportunities to practice core functions such as single-digit multiplication so that they have access to more complex concepts and procedures. Application: The Standards call for students to use math flexibly for applications in problem-solving contexts. In content areas outside of math, particularly science, students are given the opportunity to use math to make meaning of and access content. For a printable version of the CCSS Shifts in Mathematics, click the link below.Download the Shifts Full-text of the K-8 and high school Standards. Interactive version available here. The Introduction to the Math Shifts Module is a 1–4 hour module that provides you with an introduction to the key shifts required by the Common Core State Standards for Mathematics (CCSSM). CCSS for Teachers: The Shifts, is a collection of iTunes U courses designed by classroom teachers in partnership with the CCSSO, that explain the Shifts in ELA / Literacy and mathematics required by the Common Core. Brief article gives an overview of the shifts required by the Common Core State Standards and how to make them. Collection of brief videos introducing the Common Core State Standards.

<urn:uuid:8288bb3f-caa5-42e1-9498-6b0c714b63c3>

2014-10-22T22:29:08-04:00 I wold love to answer it, but what are you wanting to know? What x= or what y Solve for y -4x + 3y = 21 -4x + 3y = 21;   Add 4x to each side -4x + 4x + 3y = 21 + 4x;  Simplify 3y = 4x + 21; Divide each term by 3 3y/3 = 4x/3 + 21/3; Simplify y = 4/3x + 7 The y-intercept happens when x = 0 y = 4/3(0) + 7 y = 7 The x-intercept happens when y = 0 0 = 4/3x + 7; Subtract 7 from each side 0 - 7 = 4/3x + 7 - 7; Simplify -7 = 4/3x; Times each side by 3/4 -7 * 3/4 = 4/3x * 3/4; Simplify -21/4 = x I hope I helped Both. Oh ok so you want two equations?

crawl-data/CC-MAIN-2017-04/segments/1484560280763.38/warc/CC-MAIN-20170116095120-00397-ip-10-171-10-70.ec2.internal.warc.gz

You must have knowledge of current. As current is the charge per unit time. Resistance is the obstruction offered to the flow of current. Reason: As charge (electrons) flow through the wire (conductor), electrons collide with the walls of the conductor and they also collide with each other. Due to this collision, obstruction occurs and this is called resistance. Let us understand the concept of resistance with a common daily life experience: Suppose, you are walking at a place where there is no person, it means you are free to move. Now suppose you are moving at delhi or mumbai railway station, then what will happen? You will face lots of people and you will now face obstruction for your movements. This is basically resistance.

<urn:uuid:5f945b22-80d3-45a0-b851-07d7f1444acc>

5 September, 19:35 # 3x^2-x=10Find factorsDetermine zeros +1 1. 5 September, 21:14 0 Factors are (3x + 5) and (x - 2). Zeros are - 5/3 and 2. Step-by-step explanation: Step 1: Given quadratic equation 3x² - x = 10 ⇒ 3x² - x - 10 = 0 Step 2: Use factoring method (product and sum rule) to find factors and zeros. So the equation can be written as below. ⇒ 3x² - 6x + 5x - 10 = 0 ⇒ 3x (x - 2) + 5 (x - 2) = 0 ⇒ (3x + 5) (x - 2) = 0 These are the factors of the equation. Step 3: Find zeros by equating the factors to 0. ⇒ 3x + 5 = 0 and x - 2 = 0 ⇒ x = - 5/3 and 2

crawl-data/CC-MAIN-2021-25/segments/1623488519735.70/warc/CC-MAIN-20210622190124-20210622220124-00612.warc.gz

Open in App Not now Maximize sum by multiplying adjacent difference of Binary String with any digit • Last Updated : 20 Jan, 2023 Given a binary string S and a string of non-zero digits P of length N, the task is to maximize the sum using the given operations, which can be used as my times you can: • You can choose a substring of length 2 or 3. • Delete that substring after calculating the absolute difference of adjacent elements from left to right and take any single digit from the string P and multiply that with the absolute difference obtained by chosen substring. Examples: Input: N = 4, S = 1001, P = 6574 Output: 13 Explanation: For string S = 1001, choose a substring from index 1 to 2 as substring= 10. Absolute difference = |1 – 0| = 1 Multiply it with the digit 7 of string P = 7*1 = 7. Now chosen substring (10) is deleted from string (1001) and remaining string is = 01. Then, choose a substring from index 1 to 2 as substring = 01. Absolute difference = |0-1| = 1, Multiply it with the 6 of string P = 6*1 = 6. Now substring is deleted and string S is empty, and total maximum sum that can obtain is 7+6 = 13. Input: N = 4, S = 1111, P = 1221 Output: 2 Explanation: For, string S = 1111, choose a sub-array from index 1 to 3 as sub-array = 111. Absolute difference of first left two elements of chosen substring = |1-1| = 0, Now substring is = 01.(By replacing obtained abs. difference with first two elements). Now, absolute difference of first left two elements of current sub-array (01) = |0-1| = 1. Total absolute difference  for this substring is 1. Take digit 2 from P and multiply it with obtained absolute difference = 1*2= 2. Now delete this substring(111) from S(1111), Then remained S is = 1. We cannot make any operations further, Because S has single element and have not any adjacent element. Therefore, Maximum sum that can obtain is = 2. Approach: Implement the idea below to solve the problem: The problem can be solved by using Stack data structure and greedy approach to choose a digit from string P for multiplication. For knowing the exact algorithm to solve the problem see the Algorithm section below. Follow the steps to solve the problem: • Initialize a Stack (say stk) and create a counter and sum variables. • Make an ArrayList lets call it digits to store digits of string P and sort it. • Run a loop from 1 to N and do • If stk is empty or the top of stk = current_element, push the typecasted integer element in the stack. • Else if top of stk is not the same as current_element, pop the peek element from stk and add (1*last element of digits) into sum variable also remove used digits from the list. • Now it is possible that some element remained in the stack. Therefore, run a while loop till the stk has no element in it. Count the number of 1 remaining in the stk in the counter variable. • Modify the counter variable’s value now with (counter/3). The reason is that 3 ones can make absolute difference as 1, same as the input 2 in the example above. • Do the same, add (1*last element of digits ArrayList) into sum variable also remove used digits from the list, while counter is greater than 0. • Print the value of the sum variable. Below is the implementation of the above approach. C++ `// C++ code to implement the approach.`   `#include ` `using` `namespace` `std;` `// Function to find the maximum possible sum` `int` `maxSum(string S, string P, ``int` `N)` `{` `    ``// List to store digits of string P` `    ``vector<``int``> digits;`   `    ``// Loop for initializing digits of` `    ``// string P into list` `    ``for` `(``int` `i = 0; i < P.length(); i++) {`   `        ``// Initializing digits by` `        ``// typecasting it from string` `        ``// to integer`   `        ``digits.push_back(P[i] - 48);` `        ``// digits.add(Integer.parseInt("" + P.charAt(i)));` `    ``}`   `    ``// Sorting digits of list so that` `    ``// we can get max element at last` `    ``// index each time using greedy` `    ``// approach` `    ``sort(digits.begin(), digits.end());` `    ``// digits.sort(null);`   `    ``int` `sum = 0;` `    ``stack<``int``> s;`   `    ``// Loop for traversing on string` `    ``for` `(``int` `i = 0; i < N; i++) {`   `        ``// Creating current character` `        ``// as string so that in-built` `        ``// typecasted function can be` `        ``// apply to below string str1` `        ``string str1 = ``""``;` `        ``str1 += S[i];`   `        ``// Typecasted value of string` `        ``int` `typecasted_int = stoi(str1);`   `        ``// Condition when stack is empty` `        ``if` `(s.empty()) {` `            ``s.push(typecasted_int);` `        ``}`   `        ``// Condition when peek element` `        ``// of stack is equal to current` `        ``// typecasted element` `        ``else` `if` `(s.top() == typecasted_int) {` `            ``s.push(typecasted_int);` `        ``}`   `        ``// Condition when peek element` `        ``// is not equal to current` `        ``// typecasted element` `        ``else` `if` `(s.top() != typecasted_int) {` `            ``sum += digits.back();`   `            ``// Removing used digit from list` `            ``digits.pop_back();` `            ``s.pop();` `        ``}` `    ``}`   `    ``int` `counter = 0;`   `    ``// While loop will run till the` `    ``// stack is not empty` `    ``while` `(!s.empty()) {`   `        ``// If peek element found` `        ``// to be equal to 1` `        ``if` `(s.top() == 1) {` `            ``counter++;` `        ``}` `        ``s.pop();` `    ``}`   `    ``// Three 111 can make abs` `    ``// difference 1 Therefore total` `    ``// number of 1 are divided by 3.` `    ``counter = counter / 3;`   `    ``// Adding (abs diff.)*digit by` `    ``// getting abs difference from` `    ``// counter variable` `    ``while` `(counter-- > 0) {` `        ``sum += digits.back();`   `        ``// Removing used digit from list` `        ``digits.pop_back();` `    ``}` `    ``return` `sum;` `}`   `// Driver code` `int` `main()` `{` `    ``string S = ``"1001"``;` `    ``string P = ``"6574"``;` `    ``int` `N = S.size();`   `    ``// Function call` `    ``cout << (maxSum(S, P, N));` `}`   `// This code is contributed by garg28harsh.` Java `// Java code to implement the approach.`   `import` `java.io.*;` `import` `java.lang.*;` `import` `java.util.*;`   `class` `GFG {`   `    ``// Function to find the maximum possible sum` `    ``static` `long` `maxSum(String S, String P, ``int` `N)` `    ``{` `        ``// List to store digits of string P` `        ``ArrayList digits = ``new` `ArrayList<>();`   `        ``// Loop for initializing digits of` `        ``// string P into list` `        ``for` `(``int` `i = ``0``; i < P.length(); i++) {`   `            ``// Initializing digits by` `            ``// typecasting it from string` `            ``// to integer` `            ``digits.add(Integer.parseInt(``""` `+ P.charAt(i)));` `        ``}`   `        ``// Sorting digits of list so that` `        ``// we can get max element at last` `        ``// index each time using greedy` `        ``// approach` `        ``digits.sort(``null``);`   `        ``long` `sum = ``0``;` `        ``Stack s = ``new` `Stack<>();`   `        ``// Loop for traversing on string` `        ``for` `(``int` `i = ``0``; i < N; i++) {`   `            ``// Creating current character` `            ``// as string so that in-built` `            ``// typecasted function can be` `            ``// apply to below string str1` `            ``String str1 = ``""` `+ S.charAt(i);`   `            ``// Typecasted value of string` `            ``Integer typecasted_int = Integer.parseInt(str1);`   `            ``// Condition when stack is empty` `            ``if` `(s.isEmpty()) {` `                ``s.push(typecasted_int);` `            ``}`   `            ``// Condition when peek element` `            ``// of stack is equal to current` `            ``// typecasted element` `            ``else` `if` `(s.peek() == typecasted_int) {` `                ``s.push(typecasted_int);` `            ``}`   `            ``// Condition when peek element` `            ``// is not equal to current` `            ``// typecasted element` `            ``else` `if` `(s.peek() != typecasted_int) {` `                ``sum += ``1` `                       ``* (``int``)(digits.get(digits.size()` `                                          ``- ``1``));`   `                ``// Removing used digit from list` `                ``digits.remove(` `                    ``digits.get(digits.size() - ``1``));` `                ``s.pop();` `            ``}` `        ``}`   `        ``int` `counter = ``0``;`   `        ``// While loop will run till the` `        ``// stack is not empty` `        ``while` `(!s.isEmpty()) {`   `            ``// If peek element found` `            ``// to be equal to 1` `            ``if` `(s.peek() == ``1``) {` `                ``counter++;` `            ``}` `            ``s.pop();` `        ``}`   `        ``// Three 111 can make abs` `        ``// difference 1 Therefore total` `        ``// number of 1 are divided by 3.` `        ``counter = counter / ``3``;`   `        ``// Adding (abs diff.)*digit by` `        ``// getting abs difference from` `        ``// counter variable` `        ``while` `(counter-- > ``0``) {` `            ``sum += ``1` `* digits.get(digits.size() - ``1``);`   `            ``// Removing used digit from list` `            ``digits.remove(digits.get(digits.size() - ``1``));` `        ``}` `        ``return` `sum;` `    ``}`   `    ``// Driver code` `    ``public` `static` `void` `main(String args[])` `    ``{` `        ``String S = ``"1001"``;` `        ``String P = ``"6574"``;` `        ``int` `N = S.length();`   `        ``// Function call` `        ``System.out.println(maxSum(S, P, N));` `    ``}` `}` Python `from` `typing ``import` `List`   `# Function to find the maximum possible sum` `def` `maxSum(S, P, N):` `    ``# List to store digits of string P` `    ``digits ``=` `[]`   `    ``# Loop for initializing digits of` `    ``# string P into list` `    ``for` `i ``in` `range``(``len``(P)):` `        ``# Initializing digits by` `        ``# typecasting it from string to integer` `        ``digits.append(``int``(P[i]))`   `    ``# Sorting digits of list so that` `    ``# we can get max element at last` `    ``# index each time using greedy` `    ``# approach` `    ``digits.sort()`   `    ``sum` `=` `0` `    ``s ``=` `[]` `    ``# Loop for traversing on string` `    ``for` `i ``in` `range``(N):` `        ``# Creating current character` `        ``# as string so that in-built` `        ``# typecasted function can be` `        ``# apply to below string str1` `        ``typecasted_int ``=` `int``(S[i])`   `        ``# Condition when stack is empty` `        ``if` `not` `s:` `            ``s.append(typecasted_int)` `        ``# Condition when peek element` `        ``# of stack is equal to current` `        ``# typecasted element` `        ``elif` `s[``-``1``] ``=``=` `typecasted_int:` `            ``s.append(typecasted_int)` `        ``# Condition when peek element` `        ``# is not equal to current` `        ``# typecasted element` `        ``elif` `s[``-``1``] !``=` `typecasted_int:` `            ``sum` `+``=` `digits[``-``1``]` `            ``# Removing used digit from list` `            ``del` `digits[``-``1``]` `            ``del` `s[``-``1``]`   `    ``counter ``=` `0` `    ``# While loop will run till the` `    ``# stack is not empty` `    ``while` `s:` `        ``# If peek element found` `        ``# to be equal to 1` `        ``if` `s[``-``1``] ``=``=` `1``:` `            ``counter ``+``=` `1` `        ``del` `s[``-``1``]` `    ``# Three 111 can make abs` `    ``# difference 1 Therefore total` `    ``# number of 1 are divided by 3.` `    ``counter ``=` `counter ``/``/` `3` `    ``# Adding (abs diff.)*digit by` `    ``# getting abs difference from` `    ``# counter variable` `    ``while` `counter > ``0``:` `        ``sum` `+``=` `digits[``-``1``]` `        ``# Removing used digit from list` `        ``del` `digits[``-``1``]` `        ``counter ``-``=` `1` `    ``return` `sum`   `# Driver code` `if` `__name__ ``=``=` `"__main__"``:` `    ``S ``=` `"1001"` `    ``P ``=` `"6574"` `    ``N ``=` `len``(S)`   `    ``# Function call` `    ``print``(maxSum(S, P, N))` C# `// C# code to implement the approach.`   `using` `System;` `using` `System.Collections;` `using` `System.Collections.Generic;`   `public` `class` `GFG {`   `  ``// Function to find the maximum possible sum` `  ``static` `long` `maxSum(``string` `S, ``string` `P, ``int` `N)` `  ``{` `    ``// List to store digits of string P` `    ``ArrayList digits = ``new` `ArrayList();`   `    ``// Loop for initializing digits of` `    ``// string P into list` `    ``for` `(``int` `i = 0; i < P.Length; i++) {`   `      ``// Initializing digits by` `      ``// typecasting it from string` `      ``// to integer` `      ``digits.Add(``int``.Parse(``""` `+ P[i]));` `    ``}`   `    ``// Sorting digits of list so that` `    ``// we can get max element at last` `    ``// index each time using greedy` `    ``// approach` `    ``digits.Sort(``null``);`   `    ``long` `sum = 0;` `    ``Stack s = ``new` `Stack();`   `    ``// Loop for traversing on string` `    ``for` `(``int` `i = 0; i < N; i++) {`   `      ``// Creating current character` `      ``// as string so that in-built` `      ``// typecasted function can be` `      ``// apply to below string str1` `      ``string` `str1 = ``""` `+ S[i];`   `      ``// Typecasted value of string` `      ``int` `typecasted_int = ``int``.Parse(str1);`   `      ``// Condition when stack is empty` `      ``if` `(s.Count == 0) {` `        ``s.Push(typecasted_int);` `      ``}`   `      ``// Condition when peek element` `      ``// of stack is equal to current` `      ``// typecasted element` `      ``else` `if` `((``int``)s.Peek() == typecasted_int) {` `        ``s.Push(typecasted_int);` `      ``}`   `      ``// Condition when peek element` `      ``// is not equal to current` `      ``// typecasted element` `      ``else` `if` `((``int``)s.Peek() != typecasted_int) {` `        ``sum += 1 * (``int``)(digits[digits.Count - 1]);`   `        ``// Removing used digit from list` `        ``digits.Remove(digits[digits.Count - 1]);` `        ``s.Pop();` `      ``}` `    ``}`   `    ``int` `counter = 0;`   `    ``// While loop will run till the` `    ``// stack is not empty` `    ``while` `(s.Count != 0) {`   `      ``// If peek element found` `      ``// to be equal to 1` `      ``if` `((``int``)s.Peek() == 1) {` `        ``counter++;` `      ``}` `      ``s.Pop();` `    ``}`   `    ``// Three 111 can make abs` `    ``// difference 1 Therefore total` `    ``// number of 1 are divided by 3.` `    ``counter = counter / 3;`   `    ``// Adding (abs diff.)*digit by` `    ``// getting abs difference from` `    ``// counter variable` `    ``while` `(counter-- > 0) {` `      ``sum += 1 * (``int``)digits[digits.Count - 1];`   `      ``// Removing used digit from list` `      ``digits.Remove(digits[digits.Count - 1]);` `    ``}` `    ``return` `sum;` `  ``}`   `  ``static` `public` `void` `Main()` `  ``{`   `    ``// Code` `    ``string` `S = ``"1001"``;` `    ``string` `P = ``"6574"``;` `    ``int` `N = S.Length;`   `    ``// Function call` `    ``Console.WriteLine(maxSum(S, P, N));` `  ``}` `}`   `// This code is contributed by lokesh.` Javascript `// Javascript code to implement the approach.`   `// Function to find the maximum possible sum` `function` `maxSum(S, P, N) {` `    ``// List to store digits of string P` `    ``let digits = [];`   `    ``// Loop for initializing digits of` `    ``// string P into list` `    ``for` `(let i = 0; i < P.length; i++) {`   `        ``// Initializing digits by` `        ``// typecasting it from string` `        ``// to integer` `        ``digits.push(parseInt(``""` `+ P[i]));` `    ``}`   `    ``// Sorting digits of list so that` `    ``// we can get max element at last` `    ``// index each time using greedy` `    ``// approach` `    ``digits.sort();`   `    ``let sum = 0;` `    ``let s = [];`   `    ``// Loop for traversing on string` `    ``for` `(let i = 0; i < N; i++) {`   `        ``// Creating current character` `        ``// as string so that in-built` `        ``// typecasted function can be` `        ``// apply to below string str1` `        ``let str1 = ``""` `+ S[i];`   `        ``// Typecasted value of string` `        ``let typecasted_int = parseInt(str1);`   `        ``// Condition when stack is empty` `        ``if` `(s.length == 0) {` `            ``s.push(typecasted_int);` `        ``}`   `        ``// Condition when peek element` `        ``// of stack is equal to current` `        ``// typecasted element` `        ``else` `if` `(s[S.length - 1] == typecasted_int) {` `            ``s.push(typecasted_int);` `        ``}`   `        ``// Condition when peek element` `        ``// is not equal to current` `        ``// typecasted element` `        ``else` `if` `(s[s.length - 1] != typecasted_int) {` `            ``sum += 1` `                ``* parseInt(digits[digits.length - 1]);`   `            ``// Removing used digit from list` `            ``digits.pop();` `            ``s.pop();` `        ``}` `    ``}`   `    ``let counter = 0;`   `    ``// While loop will run till the` `    ``// stack is not empty` `    ``while` `(s.length != 0) {`   `        ``// If peek element found` `        ``// to be equal to 1` `        ``if` `(s[S.length - 1] == 1) {` `            ``counter++;` `        ``}` `        ``s.pop();` `    ``}`   `    ``// Three 111 can make abs` `    ``// difference 1 Therefore total` `    ``// number of 1 are divided by 3.` `    ``counter = counter / 3;`   `    ``// Adding (abs diff.)*digit by` `    ``// getting abs difference from` `    ``// counter variable` `    ``while` `(counter-- > 0) {` `        ``sum += 1 * digits[digits.length - 1];`   `        ``// Removing used digit from list` `        ``digits.pop();` `    ``}` `    ``return` `sum;` `}`   `// Driver code` `let S = ``"1001"``;` `let P = ``"6574"``;` `let N = S.length;`   `// Function call` `console.log(maxSum(S, P, N));`   `// This code is contributed by ksam24000` Output `13` Time Complexity: O(N * log(N)), because sorting is performed. Auxiliary Space: O(N), as list is required to store digits. My Personal Notes arrow_drop_up Related Articles

crawl-data/CC-MAIN-2023-14/segments/1679296943555.25/warc/CC-MAIN-20230320175948-20230320205948-00423.warc.gz

Posted on December 28, 2016 History of the STOP sign. Yellow 1924–1954 stop sign. Mounting height is typical. Stop signs originated in Michigan in 1915. The first ones had black lettering on a white background and were 24 by 24 inches (61 cm × 61 cm), somewhat smaller than the current sign. As stop signs became more widespread, a committee supported by the American Association of State Highway Officials (AASHO) met in 1922 to standardize them, and selected the octagonal shape that has been used in the United States ever since. The unique eight-sided shape of the sign allows drivers facing the back of the sign to identify that oncoming drivers have a stop sign and prevent confusion with other traffic signs. It was also chosen so that it could be identified easily at night, since the original signs were not reflective. The National Conference on Street and Highway Safety (NCSHS), a group competing with AASHTO, advocated a smaller pink-on-yellow stop sign. These two organizations eventually merged to form the Joint Committee on Uniform Traffic Control Devices, which in 1935 published the first Manual on Uniform Traffic Control Devices for Streets and Highways (MUTCD) detailing the stop sign's specifications. The MUTCD stop sign specifications were altered eight times between 1935 and 1971, mostly dealing with its reflectorization and its mounting height. From 1924 to 1954, stop signs were made with a black stop legend on a yellow field. In 1954, the sign gained its current white legend/red field color configuration. Red signifies stop on traffic signals, so this specification unified red as a stop signal whether indicated by sign or by light. The mounting height reached its current level of 7 ft (2.13 m) in 1971; previously, stop signs were typically mounted 2–3 feet (0.61–0.91 m)[vague] above the ground. All prices are in USD.

<urn:uuid:0c368034-80a1-4bc2-9024-e440f014d62d>

What Is It? A meningioma is one of the most common primary brain tumors. It is almost always a benign tumor that arises from cells of the arachnoid membrane. The arachnoid membrane is one of the three layers covering the brain and spinal cord. It is a thin, filmy, spider web-like tissue (hence the name arachnoid, or "spider web-like"). Therefore, these tumors actually arise outside the brain itself, but because of their close proximity to nervous system structures, they are often touching or pushing into parts of the brain or spinal cord. They can occur anywhere where arachnoid is found, throughout the intracranial compartment and the spinal column. Rarely they can be found inside the cerebral ventricles, the fluid spaces within the brain. While the large majority of meningiomas are benign and very slow growing, rarely they can have more malignant pathology and therefore act more aggressively and invasively. Most meningiomas are named for their site of origin. For example, they can be classified as convexity, parafalcine, skull base, olfactory grove, tuberculum sellae, dorsum sella, clival, petrosal, cerebellopontine, foramen magnum or spinal, to name a few, all referring to various anatomical locations around the central nervous system. What Types of Symptoms Are Typical? When small, most meningioma are asymptomatic. Depending on their location, as they enlarge, they can start to invade or cause compression of neurological structures. As this happens, symptoms related to that neurological structure can occur. Typically this progression is slow as most of these tumors grow slowly over years or decades. Neurological symptoms depend completely on the location of the tumor. Large convexity tumors may cause weakness or sensory symptoms on the opposite side of the body. Large frontal or temporal tumors sometimes cause psychological disturbances, including personality changes. Tumors of the skull base can affect the function of cranial nerves, producing problems with smell, vision, hearing and balance, facial movement, facial sensation, swallowing, and eye movements, to name just a few possibilities. In addition to these symptoms which are specific to the location of a given tumor, headache is a common finding in many of these patients. Seizure can also occur, particularly with tumors that compress parts of the cerebral cortex. How Is The Diagnosis Typically Made? After presenting for a neurological examination, generally a CT scan or MRI scan will reveal the tumor. Meningiomas have a very characteristic appearance but they can be mistaken for other tumors depending on their location. Therefore, like most other tumors, a piece of tumor tissue is needed to make a definitive diagnosis. This is usually accomplished through surgery or a biopsy. What Are Some Common Treatments? The treatment plan for these tumors varies considerably from patient to patient, depending on symptoms, and location and size of the tumor. Asymptomatic tumors may be watched closely unless the patient prefers early surgery. In the case of symptomatic tumors, surgery to remove or de-bulk the tumor is often recommended. The specific surgical approach varies depending on the tumor size and location. The goal of surgery is to remove the entire tumor, which gives the best chance for a cure or long-term control. However, in some locations complete removal is not possible or risky and therefore some tumor is left. Radiation treatments, particularly focused beam radiation or stereotactic radiation treatments, are also sometimes used. They are often effective at slowing or stopping the growth of the tumor, although some continue to grow or do not decrease in size. This is frequently used for any residual tumor which is left after surgery to delay or prevent its re-growth. As with all medical conditions, the appropriate treatment plan varies from patient to patient dependent on many factors. Each patient should discus their individual case and their options with their physician team. Return to the Brain Tumor page from the Meningioma page. Return to the Nervous System Diseases home page. This site is not intended to offer medical advice. Every patient is different, and only your personal physician can help to counsel you about what is best for your situation. What we offer is general reference information about various disorders and treatments for your education.

<urn:uuid:0758b658-fdff-4700-84e4-86284bef438d>

It’s not every day that you come across a living member of an extinct species. Nathan Whelan, a doctoral student at the University of Alabama, had such a day in 2011, when he found specimens of Leptoxis compacta on the banks of the Cahaba River. The last recorded collection of L. compacta more commonly known as the oblong rocksnail, dates back to 1933; the species was formally declared extinct in 2000. The first picture above is Figure 4 from the manuscript, published just last month. Whelan et al. conducted several tests to confirm that this species above was indeed L. compacta. They compared the shells they found with shells of other known gastropods in the same area. The shells were dissimilar in both pigmentation and pattern. The researchers also compared their findings with archival L. compacta. By using a scanning electron microscope (SEM), they found strong evidence to suggest that they had rediscovered a heretofore “extinct” species. Citation: Whelan NV, Johnson PD, Harris PM (2012) Rediscovery of Leptoxis compacta (Anthony, 1854) (Gastropoda: Cerithioidea: Pleuroceridae). PLoS ONE 7(8): e42499. doi:10.1371/journal.pone.0042499

<urn:uuid:f5a429df-1d9e-40b8-b759-49c713123786>

# How to know that $(z-z_0^3)(z-z_0^5)(z-z_0^7) = \sum_{k=0}^3 z^k z^{3-k}_0$ How to know that $$(z-z_0^3)(z-z_0^5)(z-z_0^7) = \sum_{k=0}^3 z^k z^{3-k}_0$$ with $z_0$ a root of $z^4+1$. I can check that it is true, but is there a way to tell, by seeing the LHS expression, that it can be in the RHS form ? • From $z_0^4=-1$, you have $z_0^5=z_0$, $z_0^7=z_0^3$ etc. Commented Apr 16, 2014 at 13:33 If $\omega$ is a primitive $2k$-th root of unity, then the solutions of $z^k+1 = 0$ are $\omega^{2n+1},\, 0 \leqslant n < k$, so $$z^k + 1 = \prod_{n=0}^{k-1} (z-\omega^{2n+1}),$$ \begin{align} \prod_{n=1}^{k-1} (z-\omega^{2n+1}) &= \frac{z^k+1}{z-\omega}\\ &= \frac{z^k - \omega^k}{z-\omega}\\ &= \sum_{m=0}^{k-1} z^{k-1-m}\omega^{m}\\ &= \sum_{m=0}^{k-1} z^m\omega^{k-1-m}. \end{align} So the "trick" is to recognise that $z_0^3,z_0^5,z_0^7$ are just the fourth roots of $-1$ that are different from $z_0$.

crawl-data/CC-MAIN-2024-38/segments/1725701419169.94/warc/CC-MAIN-20240920154713-20240920184713-00461.warc.gz

# Find the quotient of a polynomial division, in a given point Given $P(x)=99(x^{101}-1)-101(x^{99}-1)$, find $Q(1)$ where $Q$ is the quotient of the division between $P(x)$ and $(x-1)^3$. Obviously I could just divide the polynomials but that is not the solution I want. It is possible to figure out $Q(1)$ without doing long division, and that is the answer I am interested in. • I'm not quite clear on the question, actually. $P(x)=(x-1)^3Q(x)+R(x)$ where $R(x)$ is quadratic. To obtain $Q(1)$ differentiate each side three times with respect to $x$. $R$ vanishes and the derivatives of $Q$ get multiplied by zero when you put $x=1$. But is this what is intended? – Mark Bennet Aug 28 '17 at 21:18 If we write: $$P(x) = (x-1)^3Q(x)+ax^2+bx+c$$ then $$P'''(x) =6Q(x)+(x-1)[......]$$ On the other hand we have $$P'''(x)= 99\cdot 101 (100\cdot 99x^{98}-98\cdot 97x^{96})$$ So $$6Q(1) = P'''(1) = 99\cdot 101 (100\cdot 99-98\cdot 97)$$

crawl-data/CC-MAIN-2019-26/segments/1560627998605.33/warc/CC-MAIN-20190618023245-20190618045245-00216.warc.gz

### Present Remotely Send the link below via email or IM Present to your audience • Invited audience members will follow you as you navigate and present • People invited to a presentation do not need a Prezi account • This link expires 10 minutes after you close the presentation • A maximum of 30 users can follow your presentation Do you really want to delete this prezi? Neither you, nor the coeditors you shared it with will be able to recover it again. # Sampling Distributions - AP Stats Chapter 18 Part 2 Means by ## Steve Mays on 8 January 2013 Report abuse #### Transcript of Sampling Distributions - AP Stats Chapter 18 Part 2 Sampling Distributions - Means AP Stats - Chapter 18 Part 2 Up to this point in the chapter we have been working with sampling distributions for proportions. Well, the concepts of sampling distributions works for sample means as well. The Central Limit Theorem There are two properties of "The Central Limit Theorem" that we need to remember in order to use it. The mean of a sampling distribution will be equal to the population mean. A sampling distribution can be approximated by a Normal model. Just like with sampling distributions for proportions, when working with sample MEANS, you need to define the mean of your Normal model and the standard deviation of your Normal model. The mean of the Normal model for x-bar is equal to the population mean. The standard deviation of the sampling distribution for means is . . . Try this . . . Human gestation times have a mean of about 266 days, with a standard deviation of about 16 days. Suppose we look at the average gestation times for a sample of 100 women. if we imagined all the possible random samples of 100 women we could take and looked at the histogram of all the sample means . . . 1. what shape would it have? 2. where would the center of that histogram be? 3. what would be the standard deviation of that histogram? 1. It would be Normal shaped. 2. The center would be at 266 days. 3. SD = 16/sqrt 100 = 1.6 days The Assumptions and Conditions you must check to use a Sampling Distribution for Means are the similar as those for proportions, but just a little bit different. Randomization Condition - Same Independence Assumption or 10% Condition - You must assume that the observations are independent, but if you cannot make that assumption, you must check the 10% condition. Large Enough Condition - We will talk about this condition more in Chapter 24, but for now we will just assume that our sample is large enough. Now let's look at an example problem One Last Topic We don't always have the population standard deviation or the population proportion. If we don't have them, then we can use the sample proportion or sample standard deviation to estimate the standard deviation of the sampling distribution. This is called STANDARD ERROR. Read pgs 424 and 425 for more information. Just for fun . . . See you in stats class. Full transcript

crawl-data/CC-MAIN-2018-43/segments/1539583510893.26/warc/CC-MAIN-20181016221847-20181017003347-00324.warc.gz

# How do you find the volume of a mole of CO2? Contents The volume of one mole of CO2 produced is 24 dm^3 at room temperature and pressure. Alternatively, if your reaction took place at standard temperature and pressure (273 K, 1 atm), then the molar volume is 22.4 dm^3. ## What is the volume of 1 mole of CO2 at STP? 1 mole of CO2 (or any gas) occupies 22.4 dm^3 at s.t.p, where 22.4 dm^3 is called molar volume of a gas at s.t.p. ## How do you find molar volume of CO2 at STP? The most common molar volume is the molar volume of an ideal gas at standard temperature and pressure (273 K and 1.00 atm). The molar volume is the volume occupied by 1 mol of a gas at standard temperature and pressure (STP). It can be calculated using PV = nRT. ## How do you find the volume of a mole with gas? Calculating the volume of a gas 1. Volume = amount in mol × molar volume. 2. Volume = 0.25 × 24. 3. = 6 dm 3 ## What is CO2 volume? The amount of carbon dioxide dissolved into a beer is measured in units called “volumes of CO2.” This gets a little technical, but a “volume” is the space that the CO2 would take up at a standard temperature (32° F) and pressure (one atmosphere) if removed from the beer. ## What is the volume of 1.2 moles of CO2 at STP? At STP, 1.2 moles of O2 has a volume of 27 L. ## What is the volume of 1 mole of CO2 gas? 1 mole of all the gases at S.T.P is found to acquire a volume of 22.4L. ## What gas occupies 22.4 at STP? One mole of oxygen gas occupies 22.4 l volume at STP. ## How do I calculate STP? What is standard temperature and pressure? 1. The standard temperature is equal to: 273.15 K = 0°C = 32°F ️ … 2. The standard pressure is equal to: 1 atm = 760 Torr = 760 mm Hg = 101.35 kPa. … 3. 1 mol of ideal gas in these conditions has a volume of 22.4 Liters. ## What is the mole of CO2? Mass of 1 mole (6.023 X 1023 molecules) of CO2 is about 44g. ## How do you find moles from volume at STP? It can be written as: V = nRT/P. “P” is pressure, “V” is volume, n is the number of moles of a gas, “R” is the molar gas constant and “T” is temperature. Record the molar gas constant “R”. R = 8.314472 J/mole x K. ## How do you find moles with STP? Molar volume at STP can be used to convert from moles to gas volume and from gas volume to moles. The equality of 1mol=22.4L is the basis for the conversion factor. Many metals react with acids to produce hydrogen gas. A certain reaction produces 86.5L of hydrogen gas at STP. ## What is the volume of 1 mole of gas at STP? What is the volume of 1 mole of an ideal gas at STP (Standard Temperature and Pressure = 0 °C, 1 atm)? So, the volume of an ideal gas is 22.41 L/mol at STP. This, 22.4 L, is probably the most remembered and least useful number in chemistry. ## What is the mole formula? Avogadro’s number is a very important relationship to remember: 1 mole = 6.022×1023 6.022 × 10 23 atoms, molecules, protons, etc. To convert from moles to atoms, multiply the molar amount by Avogadro’s number. To convert from atoms to moles, divide the atom amount by Avogadro’s number (or multiply by its reciprocal).

crawl-data/CC-MAIN-2022-21/segments/1652663039492.94/warc/CC-MAIN-20220529041832-20220529071832-00144.warc.gz

# How To Calculate Mean In Excel ## Key Takeaway: • Mean is a statistical measure that represents the central tendency of a dataset. • There are different types of mean, including arithmetic mean, geometric mean, and harmonic mean. • To calculate mean in Excel, you can use the AVERAGE function, which calculates the arithmetic mean of a set of numbers. You can also use the SUM function and the COUNT function to find the sum and number of values, respectively, and then calculate the mean by dividing the sum by the count. • If you need to calculate a weighted mean, which takes into account the importance of each value, you can use the SUMPRODUCT function or the AVERAGEIF function. • To calculate a grouped mean, which is the average of a set of values that fall within specific ranges, you can use the AVERAGEIFS function or the SUMPRODUCT function. Do you struggle to compute mean in Excel? With this article, you’ll learn how to quickly and accurately calculate mean in Excel for your data analysis. Get ready to master the basics and level up your data analytics skills! ## Understanding Mean Calculations Do you use Excel? If so, you might have used mean calculations on big data sets. There are various types of means that you can calculate in Excel. It’s key to understand the differences. In this section, we’ll explore the concept of mean calculations in Excel. Firstly, we’ll define what the mean is and why it’s important. Then, learn about the different types of means that can be calculated in Excel. Finally, find out how to apply them to your data. Let’s begin to unravel the complexities of Excel mean calculations! Image credits: pixelatedworks.com by David Washington ### What is Mean? Understanding the mean is easy with this 3-step guide: 1. Work out the total of all values in the set. 2. Count how many numbers are in the dataset. 3. Divide the total by the count to get the mean. Mean is very useful when studying large sets of data. For example, to work out the amount someone spends on groceries each month, their grocery bills can be added up and the mean calculated to get an idea of the monthly spending. Mean may not be the best way to measure central tendency if there are extreme outliers or skewed data points. The same applies if you’re working with non-numeric data, such as categories or qualitative observations from surveys. When using Excel to calculate mean for larger datasets, make sure you select cells with numbers only; else calculation results may be unexpected. Now let’s dive deeper into Different Types of Mean! ### Different Types of Mean Calculations using mean are important in statistics and data analysis. When it comes to finding average values, there are different types of mean that can be used. Here’s a guide on the types of mean. 1. Arithmetic Mean: This is a commonly used type of mean calculation. It is done by adding up all the values in your sample or population and dividing the total by the number of values. 2. Geometric Mean: This type of mean calculation is suited for datasets with ratios or rates, such as financial returns calculations or interest rates. 3. Harmonic Mean: This mean is used for variable speeds like calculating an average speed over some time period. Arithmetic, geometric, and harmonic means take into account every value in your sample or population. There are other means such as modal and median averages, which have their own uses. If the data is unevenly distributed, geometric or harmonic means may work better than arithmetic means. Choosing a suitable average always depends on the end goal and understanding which one to use will help get a better picture. Next, we look at calculating means using Excel formulas- stay tuned! ## How to Calculate Mean in Excel Working with data in Excel often requires calculating the mean value. This guide will explain three methods to do so. 1. First, we’ll look at using the AVERAGE function. It’s easy and simple. 2. Second, we’ll show you how to use the SUM function, which gives more options for different types of data. 3. Lastly, we’ll go over the COUNT function. It can help you modify the mean calculation by ignoring blank cells or ones with errors. Now, let’s get started on calculating mean in Excel! Image credits: pixelatedworks.com by Adam Woodhock ### Using AVERAGE Function to Calculate Mean Using AVERAGE Function to Calculate Mean can be an easy way to get averages in Excel. Though, it may seem difficult at first, but with practice it can become simpler. Remember, this function only calculates numeric values. Text or blank cells in your data range won’t be included in the calculation. Moreover, large datasets might cause some rounding errors. My friend once needed to calculate averages from a huge amount of data using manual methods. After hours of trying with no good result, I suggested Using AVERAGE Function to Calculate Mean and it worked! Also, another great way of calculating means in Excel is Calculating Mean with SUM Function. ### Calculating Mean with SUM Function Using SUM Function is a great way to quickly calculate the mean or average of data entered into an Excel Spreadsheet. To do this, first select the cell for the results and enter ‘=SUM(“Your Data Range”)’. Replace “Your Data Range” with where your data is located. Then divide that sum by the number of cells containing data. Type “/COUNT (“Your Data Range”)”. Press “Enter”, and you’ve got the mean. Calculating mean with SUM Function has been around since the early 90s. Mobile phones then had this feature pre-installed. Another option is to use COUNT Function to Find Mean. This helps find count without including empty cells. However, it’s more complex than using sum functions in Excel. ### Using COUNT Function to Find Mean To calculate the mean in Excel, there are several functions available, such as AVERAGE, SUM, and COUNT. To use COUNT to find the mean, follow these steps: 1. Arrange the data in columns or rows. Select the cell to display the mean. 2. Enter =SUM in the selected cell and specify the range of cells containing the data in parentheses. 3. Divide by =COUNT and include the cell range again in parentheses. 4. Click enter and see the result. 5. Format it as desired. Using COUNT with SUM formulas is beneficial when calculating averages from large datasets. It takes into account any missing values. Traditionally, people use formulae like “AVERAGE” or “SUMIF” for computing means in excel. But these methods can be impacted by outliers, leading to inaccurate results. Excel’s built-in functions offer more precision and less bias. To calculate Weighted Mean in Excel, one must consider criterion weights based on relevance. Multiples criteria evaluation factors apply corresponding weightings to get more-rigorous analysis using Excel statistics formulae like AVERAGEIF etcetera. ## How to Calculate Weighted Mean in Excel I’m thrilled to introduce you to a new section about finding weighted means in Excel! This useful function is essential for lots of professions such as finance, economics, education, and healthcare. We’ll look at two methods: the SUMPRODUCT function and the AVERAGEIF function. They’re both simple and efficient! I’ll show you step-by-step how to use them, so you can start using them right away. Image credits: pixelatedworks.com by Adam Woodhock ### Using SUMPRODUCT Function for Weighted Mean Calculation Calculating a weighted mean with Excel? The SUMPRODUCT function is the way to go! This formula multiplies each value by its weight, adds them together and divides the result by the total weight. This gives more significance to certain values and gives you a more meaningful average. Let’s take a look at an example. Here’s a table with four different values and their respective weights: Values Weights 50 3 60 2 70 4 80 1 To calculate the weighted mean for this data, use the formula: =SUMPRODUCT(Values,Weights)/SUM(Weights) So, for the above example, it will look like: =SUMPRODUCT(A2:A5,B2:B5)/SUM(B2:B5) =(50*3+60*2+70*4+80*1)/(3+2+4+1) =63.33 The SUMPRODUCT function helps you quickly calculate weighted means when dealing with large datasets. Don’t miss out on this step – try it and see how it improves your results! Next up, we’ll explore another useful method for calculating weighted means: using the AVERAGEIF function. ### Using AVERAGEIF Function for Weighted Mean Calculation The AVERAGEIF function in Excel makes calculating weighted mean a piece of cake! Follow these 4 steps: 1. Input your data list with two or more columns. One consists of weights and the other has values. 2. Use the AVERAGEIF function in the cell where you want the answer. Set up criteria to include only cells with non-zero weights. 3. Specify which weight column and range should be evaluated based on criteria. 4. Indicate which data column should be evaluated for the formula. AVERAGEIF Function for Weighted Mean Calculation is great for determining how much emphasis each value should have when calculating an average. It’s useful for calculating grades, where students’ performances have different weights. If it looks intimidating at first, don’t worry! Breaking it down into steps makes it easy. I used this formula for calculating my high school grades. Teachers had a weighted grading system, so each assignment’s score was given the right importance when calculating term averages or overall scores. Now let’s talk about How to Calculate Grouped Mean in Excel? ## How to Calculate Grouped Mean in Excel Data analysis often requires finding the mean or average value. But, what about grouped data? In this article, there are two ways to calculate the grouped mean in Microsoft Excel. 1. Firstly, the AVERAGEIFS function. It uses criteria to select specific data groups for calculating. 2. Secondly, the SUMPRODUCT function. It provides a versatile way to manipulate and evaluate data sets. Excel now has the tools to make data analysis more effective! Image credits: pixelatedworks.com by Joel Jones ### Using AVERAGEIFS Function for Grouped Mean Calculation Insert data groupings into your Excel sheet. Label each column with headings. Click an empty cell to display the calculated average. Type “=AVERAGEIFS“. Inside parentheses, list the range of cells with data. Follow with a comma. Enter criteria range and its value. Repeat this for all conditions. Close parentheses and hit enter. Result will display on cell. Using AVERAGEIFS Function is efficient. It saves time, especially when filters are involved. • Computes averages based on multiple criteria. • Only relevant info taken into account. • Increases productivity when dealing with large data sets. • Can streamline grouped mean calculations. • Failure to incorporate these features may lead to poor data analysis. Impact business decisions and outcomes negatively. ### Using SUMPRODUCT Function for Grouped Mean Calculation To use SUMPRODUCT Function for Grouped Mean Calculation: 1. Enter frequency and class intervals into Excel worksheet. 2. Create two columns—one for midpoints, one for frequencies. 3. Enter =SUMPRODUCT(midpoints,frequencies)/SUM(frequencies) in a new cell, substituting “midpoints” and “frequencies”. Multiplying each midpoint by its frequency and adding the products yields the grouped mean. This provides a more accurate result than calculation without grouping data. Ensure correct entry of both midpoints and frequencies. Avoid mismatching ranges which cause incorrect calculations. Reference materials and tutorials provide help if unsure. Following these steps precisely will help you obtain an accurate group mean using Excel and SUMPRODUCT Function. ## Five Facts About How to Calculate Mean in Excel: • ✅ Mean is the average of a set of numbers, calculated by adding all the numbers in the set and then dividing the sum by the total number of elements in the set. (Source: Investopedia) • ✅ The AVERAGE function in Excel is used to calculate the mean of a set of numbers. (Source: Microsoft Support) • ✅ To calculate mean in Excel, select the cell where you want to display the result, enter the formula “=AVERAGE(range)”, and replace “range” with the range of cells containing the numbers. (Source: Excel Easy) • ✅ The arithmetic mean is one type of mean, but there are also other types such as the geometric mean and the harmonic mean. (Source: Statistics How To) • ✅ Mean is a commonly used statistical measure that provides insight into the central tendency of a set of data. (Source: ThoughtCo) ## FAQs about How To Calculate Mean In Excel ### How to Calculate Mean in Excel? The mean (average) is a statistical measure which is calculated by summing up a set of values and dividing the sum by the total number of data points. Here’s how to calculate mean in Excel: 1. Select the cell where you want the mean to be displayed 2. Type in the formula “=AVERAGE(range)” where “range” refers to the cells which contain the data you want to find the mean of 3. Press enter, and the mean will be displayed in the cell you selected. ### Can you explain the AVERAGE function in Excel? The AVERAGE function calculates the mean (average) of a set of values in a given range. For example, the formula “=AVERAGE(A1:A5)” would calculate the average of the values in cells A1 through A5. ### What is the difference between AVERAGE and MEDIAN in Excel? The AVERAGE function calculates the mean of a set of values, while the MEDIAN function calculates the median (middle) value from a set of values. The mean is affected by outliers and extreme values, while the median is not. ### Is there a shortcut to calculate mean in Excel? Yes, there is a shortcut to calculate mean in Excel. You can simply select the range of cells containing the values you want to calculate the mean for, and the mean value will be displayed in the status bar at the bottom of the Excel window. ### Can I calculate mean for non-numeric values in Excel? No, you cannot calculate mean for non-numeric values in Excel. If you try to use the AVERAGE function on a range of cells containing non-numeric values, Excel will display an error message. ### Can I customize the number of decimal places displayed in the mean result? Yes, you can customize the number of decimal places displayed in the mean result. Simply right-click on the cell containing the mean result, select “Format Cells”, and choose the number of decimal places you want to display.

crawl-data/CC-MAIN-2024-10/segments/1707947475711.57/warc/CC-MAIN-20240301225031-20240302015031-00084.warc.gz

• How to get 1/6 or 3/4 of a given volume with 3 cups? • Not so difficult, right? • Now lets try 3/4. • Ok, but can you get 1/5? Why not? www.sciensation.org | Ciênsação hands-on experiments are published as Open Educational resources under a Creative Commons Attribution-ShareAlike 4.0 International License. Maths Game age: 10 – 13 # Liquid fractions A simple yet memorable way to teach fractions with 3 cups and a bit of water. Learning objective Understanding fractions. Relating mathematical concepts to real-world problems. Cups Water Food coloring (optional) Preparation Place in front of each student group 3 identical plastic cups and fill one of them up to a specific mark. Then let them work on their tasks. Have for yourself a cup or measuring cylinder where you marked the level for 1/6 and 3/4 of the water volume you filled into the students' cups. Once a group of students thinks they have solved the 'riddle', give them your reference cup to check if they really measured 1/6 or 3/4, respectively. You are only allowed to transfer water between the three cups. 1. Get 1/6 of the original volume of water in one cup. 2. Get 3/4 of the original volume of water in one cup. 3. Get 1/5 of the original volume of water in one cup. Findings Distributing the water evenly over three cups leaves each cup with 1/3 of the water. By emptying one into another and dividing the remaining 1/3 equally in two cups, you end up with two cups containing 1/6th of the original volume. Dividing the water equally between two cups and then dividing the water in one of them with the third, still empty cup, will produce two cups with 1/4 and one cup of 1/2 of the original volume. To get 3/4, pour one of the 1/4 cups in the 1/2 cup. To measure with the same method 1/5 of the volume, at least 5 cups are needed. It is not possible to do this with three cups since they only allow divisions by two or three. Although your students might find this question 'stupid', you can draw later on this experience when discussing common denominators. The goal of this experiment is to link the abstract notion of fractions and their operations to a physical experience. It is therefore important to make this relation explicit in the discussion that follows the experiment, e.g. by drawing on a whiteboards the cups with their respective fill levels and the corresponding fractions and operations (division, summation). Tags

crawl-data/CC-MAIN-2024-18/segments/1712296818711.23/warc/CC-MAIN-20240423130552-20240423160552-00453.warc.gz

# Maths and Numeracy in the Real World Whatever we do, wherever we go, we’re always using maths and numeracy without even realising it. Making students see that maths is not just something we learn at school, but something we need in the real world and for future jobs, is a really good way of engaging them more with the subject. To inspire your students to realise this, we’ve put together a number of ideas you can show your students about how they use maths in real life, as well as just some fantastic EducationCity Maths activities, which you can get quickly and easily, to use in your classroom. Let’s take a look… #### 1. Maths In the Kitchen: Baking and Cooking When we bake and cook, we use a high ‘degree’ of mathematical skill – like what we did there? Every ingredient needs to be measured out to the right amounts and as well as this, quite a lot of the time we need to multiply or divide to get the exact amounts we’re after. Lots of what you do in the kitchen requires maths – even turning on the oven! To help show your students how they use maths in baking and cooking, check out our Marvellous Muffins Learn Screen and Activity… ## Learn Screen Marvellous Muffins – A tutorial on how to read unnumbered scales. • England: Year 5 • Scotland: Second ** • Wales: Year 5 & 6 • N. Ireland: P7 ## Activity Marvellous Muffins – Read partially numbered scales and be able to work out the difference between 2 given points on them. • England: Year 5 • Scotland: Second ** • Wales: Year 5 & 6 • N. Ireland: P7 #### 2. Using Maths & Money We all need to know what to do with money – and this involves counting and percentages too. When we buy things at the local supermarket, or spend money online, we all might use adding or percentage skills to work out how much we’ve spent… or saved! So that you can show your students how we use money in the real world, why not take a look at these Activities? ## Activity Show Me the Money – Combine different coins to make the same amounts of money. • England: Year 2 • Scotland: Early *** • Wales: Year 2 • N. Ireland: P3 Help Granny – Multiplication using money. Click on the coins to answer. • England: Year 4 • Scotland: Second * • Wales: Year 4 • N. Ireland: P6 Summer Break – Solve conversion problems. • England: Year 6 • Scotland: Second ** • Wales: Year 6 & 7 #### 3. Telling the Time We’re also using maths when we tell the time – and telling the time is a great and valuable skill to have as students will be able to practise concepts like subtraction and addition. Whether you have a diary or a calendar too, you’re going to need time skills to be able to schedule in when you’re going to meet friends or go to occasions, or even know when you have important appointments and how long you have to get ready. To help you out with introducing time in the classroom, here are some EducationCity maths games to try… ## Topic Tool Time – Explore time on an analogue clock to the hour and half-hour. • England: F2 & Year 1 • Scotland: Early *, Early ** & Early *** • Wales: Reception & Year 1 • N. Ireland: P1 & P2 Time – Explore time on both a digital clock and analogue clock to the nearest 5 minutes. • England: Year 2 • Scotland: First *, First ** & First *** • Wales: Year 2 & 3 • N. Ireland: P3 & P4 ## Activity Stig and the Bus – Read the time from timetables. • England: Year 2 • Scotland: Second * • Wales: Year 5 & 6 • N. Ireland: P7 There’s so much more to explore! Just use the search bar to find even more maths games on EducationCity related to cooking, money and time.

crawl-data/CC-MAIN-2024-33/segments/1722641008125.69/warc/CC-MAIN-20240811172916-20240811202916-00676.warc.gz

If you're seeing this message, it means we're having trouble loading external resources on our website. If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked. # Intro to the Polynomial Remainder Theorem The Polynomial Remainder Theorem allows us to determine whether a linear expression is a factor of a polynomial expression easily. Check it out! ## Want to join the conversation? • What is the difference between a binomial and a polynomial? • binomial-they are with two terms polynomial-monomial,binomial,trinomial everything are considered to be a polynomial • why do i feel like him saying, "Starting polynomial long division is a good way to start your morning." Was a cry for help • I can't relate...it's past midnight for me... just squeezin' in some late-night studying :)) • When would it be useful to just calculate the remainder but not the quotient of polynomial division? Can anyone provide an example? • Is the remainder theorem only true when you're dividing by x-a? Or is it true for x+a as well? • It is true for x + a as well. x + a is another way of writing x - (-a). This comes into play when using synthetic division. Sometimes you'll be given a polynomial and a binomial in the form x + a. If it was x + 9, you would just take the opposite of 9, which is -9. Hope that helps. • Shouldn't it be f(-a): You have x-1, and then you plug in 1. No? • x-a! buddy a=1 so we plug in one!! if it were x+a than you would be right buddie • Does the polynomial remainder theorem also work on equations where the denominator 'x-a' where x has a coefficient or to anything greater than the 1st degree? • It would work when x has a coefficient but when you have a denominator or divisor that has a degree that's greater than one, the remainder theorem wouldn't work as the remainder for higher degree terms is not constant.. (I got this from another person's answer on this website) • What is the polynomial remainder theorem then? • Can you use this Theorem when dividing a polynomial with x+a, with a being some positive constant?

crawl-data/CC-MAIN-2023-06/segments/1674764500080.82/warc/CC-MAIN-20230204012622-20230204042622-00692.warc.gz

The Snowy Owl (Bubo scandiacus) is a large owl of the typical owl family Strigidae. The Snowy Owl was first classified in 1758 by Carolus Linnaeus, the Swedish naturalist who developed binomial nomenclature to classify and organize plants and animals. The bird is also known in North America as the Arctic Owl or the Great White Owl. Until recently, it was regarded as the sole member of a distinct genus, as Nyctea scandiaca, but mtDNA cytochrome b sequence data (Olsen et al. 2002) shows that it is very closely related to the horned owls in the genus Bubo. Typical female, Korkeasaari (Finland) This yellow-eyed, black billed white bird is easily recognizable. It is 53-65 cm (20-26 inches) long with a 125-150 cm (50-60 in) wingspan. Also, these birds can weigh anywhere from 1.8-3 kg (3.5-6.6 lbs). The adult male is virtually pure white, but females and young birds have some dark scalloping; the young are heavily barred, and dark spotting may even be predominate. Its thick plumage, heavily-feathered feet, and coloration render the Snowy Owl well-adapted for life north of the Arctic Circle. Snowy Owl calls are varied, but the alarm call is a barking, almost quacking krek-krek-krek-krek; the female also has a softer mewling pyee-pyee-pyee-pyee or prek-prek-prek. The song is a deep repeated gawh. They may also clap their beak in response to threats or annoyances. While called clapping, it is believed this sound may actually be a clicking of the tongue, not the beak. Young Owl on the tundra at Barrow Alaska The Snowy Owl is typically found in the northern circumpolar region, where it makes its summer home north of latitude 60 degrees north. However, it is a particularly nomadic bird, and because population fluctuations in its prey species can force it to relocate, it has been known to breed at more southerly latitudes. During the last ice age, there was a Central European paleosubspecies of this bird, Bubo scandiacus gallicus, but no modern subspecies are recognized. This species of owl nests on the ground, building a scrape on top of a mound or boulder. A site with good visibility, ready access to hunting areas, and a lack of snow is chosen. Gravel bars and abandoned eagle nests may be used. Breeding occurs in May, and depending on the amount of prey available, clutch sizes range from 5 to 14 eggs, which are laid singly, approximately every other day over the course of several days. Hatching takes place approximately five weeks after laying, and the pure white young are cared for by both parents. Both the male and the female defend the nest with their young from predators. Some individuals stay on the breeding grounds while others migrate. Snowy Owls winter south through Canada and northernmost Eurasia, with irruptions occurring further south in some years. They have been reported as far south as Texas, Georgia, the American Gulf states, southern Russia, northern China and even the Caribbean. Between 1967 and 1975, Snowy Owls bred on the remote island of Fetlar in the Shetland Isles north of Scotland, UK. Females summered as recently as 1993, but their status in the British Isles is now that of a rare winter visitor to Shetland, the Outer Hebrides and the Cairngorms. Hunting and diet This powerful bird relies primarily on lemmings and other rodents for food, but at times of low prey density, or during the ptarmigan nesting period, they may switch to juvenile ptarmigan. As opportunistic hunters, they feed on a wide variety of small mammals and birds such as meadow voles and deer mice, but will take advantage of larger prey, frequently following traplines to find food. Some of the larger mammal prey includes mice, hares, muskrats, marmots, squirrels, rabbits, prairie dogs, rats, moles, and entrapped furbearers. Birds include ptarmigan, ducks, geese, shorebirds, ring-necked pheasants, grouse, American coots, grebes, gulls, songbirds, and short-eared owls. Snowy Owls are also known to eat fish and carrion. Most of the owls' hunting is done in the "sit and wait" style; prey may be captured on the ground, in the air or fish may be snatched off the surface of bodies of water using their sharp talons. Each bird must capture roughly 7 to 12 mice per day to meet its food requirement and can eat more than 1,600 lemmings per year. Snowy Owls, like many other birds, swallow their small prey whole. Strong stomach juices digest the flesh and the indigestible bones, teeth, fur, and feathers are compacted into oval pellets that the bird regurgitates 18 to 24 hours after feeding. Regurgitation often takes place at regular perches, where dozens of pellets may be found. Biologists frequently examine these pellets to determine the quantity and types of prey the birds have eaten. When large prey are eaten in small pieces, pellets will not be produced. Though Snowy Owls have few predators, the adults are very watchful and well equipped to defend against any kind of threats towards them or their offspring. During the nesting season the owls face Arctic foxes and swift-flying jaegers and must be very careful not to leave their eggs unattended. Environmental conditions also cause local threats of food shortages, but their ability to be mobile permits them to move to areas were supplies may be more sufficient. Human activities probably pose the greatest danger to these birds, through collisions with power lines, fences, automobiles, or other structures that impose on their natural habitat. Now, Canadian provincial and territorial regulations have introduced prohibitions of killing of these birds in all parts of Canada, where they are most abundant, but the owls are still used for certain study programs. This species is an extremely important component to the food web in the tundra ecosystem and during its visits to the south, the Snowy Owl may play a useful role in the natural control of rodents in agricultural regions. In popular culture - Steve (voiced by Jonathan Katz), the therapist who helps Jimmy the penguin with his love life in Farce of the Penguins is a Snowy owl. - The O RLY? owl is a Snowy owl. - In the 1994 film, Dumb and Dumber, Lloyd Christmas (Jim Carrey) accidentally kills an Icelandic Snowy Owl with the cork of a bottle by launching it. - The Snowy Owl is the official bird of Quebec. - The Snowy Owl was depicted on the 1986 series Canadian $50 note . - A Snowy White Owl named "Gamma" is the mascot of the fraternity Phi Gamma Delta. - The Snowy Owl is a familiar in the popular online game, Kingdom of Loathing. - In the Harry Potter series by J. K. Rowling, Harry's owl, Hedwig, is a Snowy Owl. - Although female, Hedwig is played by a male owl in the Harry Potter movies because males are more thoroughly white, lacking the barring of females. - Queen Siv's servant Myrrthe from the "Guardians of Ga'Hoole" book series was a Snowy owl. So were Borron and Barran, Madame Plonk ,and The Rogue Smith of Silverveil. - A Snowy owl is the logo of the cigar brand White Owl. - The Inuit name for the Snowy Owl is "ookpik", "okpik" or "ukpik" - Okpik, a cold-weather adventure program for the Boy Scouts of America - Abe Okpik, an Inuvialuit who was instrumental in helping Inuit obtain surnames - The Snowy Owl is one of many animals featured in the 2002 game Impossible Creatures - A Snowy Owl is featured prominently on the cover painting of Canadian rock band Rush's 1975 album Fly By Night. A live owl appears on the cover of Rush's 1981 live album Exit...Stage Left, referencing Fly By Night.

<urn:uuid:8a5ab65c-114c-44d1-ac50-c45596fa74e6>

4 July, 20:18 # Calculate the specific heat of a metallic element if 50.0 g of the metal need 314 joules of heat energy to raisethe temperature from 25°C to 50°C. +3 1. 4 July, 20:25 0 c = 0.25 j/g.°C Explanation: Specific heat capacity: It is the amount of heat required to raise the temperature of one gram of substance by one degree. Formula: Q = m. c. ΔT Q = amount of heat absorbed or released m = mass of given substance c = specific heat capacity of substance ΔT = change in temperature Given dа ta: Mass of metal = 50.0 g Heat needed = 314 j Initial temperature = 25°C Final temperature = 50 °C Specific heat = ? Solution: ΔT = 50 °C - 25°C = 25°C Q = m. c. ΔT c = Q / m. ΔT c = 314 j / 50.0 g. 25°C c = 314 j / 1250 g. °C c = 0.25 j/g.°C

crawl-data/CC-MAIN-2024-30/segments/1720763514917.3/warc/CC-MAIN-20240719170235-20240719200235-00420.warc.gz

# Understanding Work Done and the Role of Force in Solving Problems | 360J Example • Nubcake In summary, the confusion arises from the use of weight and force in the calculation of work done on an object. By convention, 0J potential energy is used at ground level, and when the barrel is raised 1.8m, it gains 360J of potential energy. The person pushing the barrel does positive work, while gravity does negative work. The net work done on the object is 0, explaining why it does not gain any kinetic energy. Nubcake I have always managed to solve problem asking how much work is being done but then once I start to think about the work being done I get confused. E.g A barrel of weight 200N is raised by a vertical distance of 1.8m by being moved along a ramp. The work done here would be 200 x 1.8 =360J . I do not understand why 200N is used since it is the weight acting vertically downwards opposite the distance moved in the direction of the force. So why is work being done shouldn't work be done when the barrel is moving in the same direction as its own weight? Or is it that 200N is the force that the person is giving to move it up along the ramp hence it is in the same direction of the force.Can someone clear this up for me? When you change heights you change the barrel's potential energy. By convention we arbitrarily use 0J potential energy at ground level and when you raise it 1.8 m then you get PE = F * d = 200N * 1.8m = 360 J then if you let it fall back to the ground it have 360 J of kinetic energy of movement at ground level which it then dissipates on impact. jedishrfu said: When you change heights you change the barrel's potential energy. By convention we arbitrarily use 0J potential energy at ground level and when you raise it 1.8 m then you get PE = F * d = 200N * 1.8m = 360 J then if you let it fall back to the ground it have 360 J of kinetic energy of movement at ground level which it then dissipates on impact. I was more concerned about the direction and the force ; doesn't the weight of the barrel act in the opposite direction of movement? Nubcake said: I was more concerned about the direction and the force ; doesn't the weight of the barrel act in the opposite direction of movement? Yes. So the work done by gravity is negative (-360 J). By definition, the change in gravitational potential energy is equal to the negative of the work done by gravity. So, the change in potential energy is +360 J (potential energy increases when gravity does negative work). The work done by the person pushing the thing up the ramp is positive, since the displacement and force are in the same direction. Let's assume that there is negligible friction. Let's also assume that the object is pushed up the ramp at a constant speed (neglecting the initial acceleration to get it moving). Then the forces parallel to the ramp have to be balanced, which means that the person pushes with a force equal to mgsinθ, which is the component of the weight that acts "down the ramp." The work done is then mgsinθ*d, where d is the distance traveled along the ramp. But from basic trigonometry, d = h/sinθ, so W = mgsinθ*(h/sinθ) = mgh. The person does an amount of work equal to mgh = +360 J (but the ramp allows him to do so with a smaller force than if he just lifted it vertically). Notice that the person does +360 J of work, and gravity does -360 J of work on the object, so the NET work done on the object is 0 (which makes sense because it has 0 net force on it). This explains why it does not gain any kinetic energy. (Recall that the work-energy theorem says that the work done on an object is equal to its change in kinetic energy). What cepheid is saying is that he uses less force to push it up the ramp, but he has to push it for a greater distance. So the product of the force times the distance comes out the same as if he had just lifted it vertically with a larger force over a smaller distance. ## What is work and how is it calculated? Work is the transfer of energy that results in an object being displaced. It is calculated by multiplying the force applied to an object by the distance it is moved in the direction of the force. ## What is the role of force in solving problems involving work? Force is necessary for work to be done on an object. Without force, no energy would be transferred and no work would be done. ## How does the direction of force affect the work done on an object? The direction of force is important in determining the work done on an object. If the force is applied in the same direction as the displacement, then all of the force contributes to the work done. However, if the force is applied in a different direction, only the component of the force in the direction of the displacement contributes to the work done. ## What are some real-life examples of work being done? Some examples of work being done include lifting a book off a table, pushing a shopping cart, and pulling a wagon. Essentially, any time an object is being moved by a force, work is being done. ## How is the concept of work applied in different fields of science? In physics, work is used to understand the transfer of energy and the motion of objects. In chemistry, work is used to understand chemical reactions and the formation of compounds. In biology, work is used to understand the movement and function of living organisms. In engineering, work is used to design and create machines and structures that can perform tasks efficiently. • Mechanics Replies 24 Views 342 • Mechanics Replies 2 Views 15K • Mechanics Replies 6 Views 2K • Mechanics Replies 9 Views 964 • Mechanics Replies 6 Views 2K • Introductory Physics Homework Help Replies 2 Views 208 • Mechanics Replies 5 Views 1K • Mechanics Replies 4 Views 1K • Mechanics Replies 18 Views 1K • Mechanics Replies 4 Views 4K

crawl-data/CC-MAIN-2024-33/segments/1722640372747.5/warc/CC-MAIN-20240803153056-20240803183056-00855.warc.gz

0 # Are all teiangles have opposite sides that are parallel? Updated: 12/21/2022 Wiki User 14y ago If you mean "triangle", a triangle can never have two parallel sides. If you mean "triangle", a triangle can never have two parallel sides. If you mean "triangle", a triangle can never have two parallel sides. If you mean "triangle", a triangle can never have two parallel sides. Wiki User 14y ago Wiki User 14y ago If you mean "triangle", a triangle can never have two parallel sides. Earn +20 pts Q: Are all teiangles have opposite sides that are parallel? Submit Still have questions? Continue Learning about Math & Arithmetic Related questions A square's opposite sides are parallel and all the sides are congruent. ### Are all sides in a rectangle parallel? Yes opposite sides are parallel in a rectangle ### What is a four-sided polygon with opposite sides parallel and all sides congruent? A four-sided polygon with opposite sides parallel and all sides congruent is a square. ### What is the name of a quadrilateral with both pairs of opposite sides parallel? This is called a Parallelogram. Each pair of opposite sides will then be the same length. If all four sides are the same length, and opposite sides are parallel, then it is a rhombus. NB: If all four sides are the same length then the opposite sides must be parallel. ### What quadrilateral have opposite sides are parallel? Quadrilaterals that have parallel opposite sides (assuming that each side is parallel to its opposite): A parallelogram A rectangle A square (really a rectangle with all sides equal in length) ### Does a parallelogram have two pairs of opposite sides parallel? Yes it does. All the four sides have an opposite. The opposites are parallel so in a parallelogram there are two pairs of parallel sides. ### Is all quadrilaterals are parallelograms truth? opposite sides are parallel opposite sides congruent opposite angles are equal opposite lines parallel 1 pair opp. lines parallel and congruent ### What plane shape has all sides equal and opposite sides parallel? Square. Square is a quadrilateral and all sides are equal and opposite sides are parallel, the sum of the interior angle measure 3600. ### Are the sides of a rhombus congruent and parallel? All 4 sides are congruent and opposite sides are parallel to each other. ### What are two differences trapaziod and a rhombus has? All sides of a trapezoid cannot be congruent, all sides of a rhombus are congruent. All opposite sides of a rhombus are parallel, only 1 pair of opposite sides of a trapezoid are parallel, the other pair are cannot be parallel. ### What geometric figure is a quadrilateral has four equal sides an both pairs of opposite sides are parallel? A square and a rhombus both have all sides equal and opposite sides parallel. ### What is a quadrilateral with two sets of opposite sides which are parallel? Take your pick: square, rectangle, rhombus or parallelogram which all have opposite parallel sides

crawl-data/CC-MAIN-2024-30/segments/1720763514801.32/warc/CC-MAIN-20240717182340-20240717212340-00419.warc.gz

CSS SYLLABUS SOCIOLOGY – 100 MARKS I. General Sociology 1. Individual: Sociability or the sociality of man. 2. Culture: Meaning and Characteristics (Culture is variable, learnt, social, shared, Transmissive, dynamic and adaptive), types (Material, Non –material), functions (transfer of knowledge, define situation, provide Behaviour pattern, moulds personality) and elements of culture (norms, values, beliefs, sanctions, customs).Culture and Socialization; formal and non-formal socialization, transmission of culture, cultural relativism. Sub-cultures. Ethnocentrism and xenocentrism, Cultural lag, High culture and popular culture. Multiculturalism, assimilation, and acculturation. 3. Society: Meaning and characteristics. Community; meaning and characteristics. Individual and society. Relationship between individual and society. Two main theories regarding the relationship of man and society (i) the social contact theory and (ii) the organismic theory. Social and cultural evolution of society (Hunting and Gathering Society, Herding and Advance Herding Society, Horticultural Society, Agrarian Society, Industrial Society, Post modern Society). 4. Social Interaction: Caste and classes, Forms of social classes, Feudal system in Pakistan, Social Mobility-nature of social mobility and its determinants in Pakistani society, Culture of poverty. 5. Social Control: Mechanisms of social control-formal and informal means of social control, Anomie, Alienation and social Integration-Means of social integration in Pakistani Society. 6. Social and Cultural Change and Social Policy: Processes of Social and Cultural Change-discovery, inhibitions to social and cultural change in Pakistan Social planning and directed social and cultural change, effect of Industrialization, Urbanization, Modernization and Modern Means of Communication on Social Change. 7. Public Opinion: Formation of Public, Opinion, Concept of opinion leader, characteristics of opinion leadership. 8. Community: The rural community, Traditional Characteristics of rural life, The urban community, Rural – Urban convergence, Urbanism, Future of cities in Pakistan. 9. Social Institutions: The nature and genesis of institutions, the process of institutionalization, Functions of Social Institutions: Family, Religion, Education, Economy and Politics. 10. Social Problems in Pakistan: High population growth rate, Rural –urban migration. Issues of technical/vocational training, Deviance and street crime, Unemployment, illiteracy and School drop out, Smuggling, Prostitution, Poverty, Drug Addiction, Child Labour and Abuse Bonded Labour, Social Customs and Traditions effecting Women in Pakistan, Violence Against Women’s and Domestics Violence, Issues concerning the Elderly’s in Pakistan. II. Sociological Theory: Three sociological perspectives: Structural Functionalism, Symbolic interactions and Conflict. Theorists: Ibn-i-Khaldun Spencer, August Comte, Emile Dukheim, Max Weber, Kari Marx, Parson. III. Methods of Sociological Research: Scientific Method, Steps in research, Types of Questionnaire Research Design, Surveys, Observation and Case Studies.

<urn:uuid:19cadc6f-cd62-4e94-8004-0234c880efd1>

## Elegant proofs 2 – The area of a circle We are so familiar with the formula for the area enclosed by a circle that we tend not to think much about how it was derived, at least I don’t. The proofs of the formula are in fact many and varied; the first one found by Google is at: http://www.artofproblemsolving.com/LaTeX/Examples/AreaOfACircle.pdf Don’t worry, that’s not the elegant one. There are many proofs that don’t (directly) involve the use of calculus, and Wikipedia gives a good sample of them: http://en.wikipedia.org/wiki/Area_of_a_disk of which the rearrangement proof is perhaps the most elegant.  Another presentation of this proof is given here (along with Archimedes’ equally elegant derivation of the volume of a sphere): http://www.mathreference.com/geo,circle.html Yesterday I came across an approach that to me seems even simpler, based on a post at: http://foxmath.wordpress.com/2008/06/24/perimeter-area/ This shows that for any regular polygon with an area equal to its circumference, the length of the apothem (the red line in the diagram to the left) is 2.  This is immediately obvious from the fact that the area of each individual triangle is equal to the base length, when the height equals 2. In the limit as the number of sides of a regular polygon tends to infinity the polygon approaches a circle, and the length of the apothem approaches the radius of the enclosing circle.  It therefore follows that the area of a circle of radius 2 is equal to its circumference; i.e. 4.pi. A circle of radius R may be scaled to radius 2 by multiplying the radius by 2/R.  The radius of this circle is then 4pi x (R/2)^2 = pi.R^2. Finally a “wordless” proof provided by the people at SSSF: http://www.maa.org/pubs/Calc_articles/ma018.pdf This entry was posted in Maths, Newton and tagged , , , , . Bookmark the permalink. ### 4 Responses to Elegant proofs 2 – The area of a circle 1. mrt says: Hi – Troubled by the thought of an area equal to a circumference. Implicit in that is that units-squared equals units. Physicists would disagree. 😉 …mrt Like 2. dougaj4 says: mrt – you are quite right that an area can’t be physically equal to a length, but they can have the same numerical value, which is what the proof is concerned with. Watch out for an upcoming post on the history of dimensional analysis! 🙂 Like 3. Piyush Gupta says: Thanks a lot dude!! 😀 Like 4. SasQ says: @mrt: You’re right, the area is not the same as length, so they cannot be equated *directly*. But if `A` is area, and `L` is length, then you can make an equation like this: `A = 1*L`. Multiplying by `1` doesn’t change anything. But now you can interpret `1*L` as a *rectangle* of `1` by `L`, which has the area of `L`. So you are now comparing areas with areas, and this is OK. To better understand this concept, you can see that any length is a multiple of a unit length, called `1`. So `L` is really `L*1` (read it: “`L` units of length”). Similarly, an area is a multiple of a unit area, which is a `1` by `1` square. So when you say that something has an area `A`, what you really say is this: `A * 1*1` (read it: “`A` units of area, that is, those little 1×1 squares”). Can you see now how can you jump between lenghts and areas and volumes etc. and why? It’s all about units and the fact that you can multiply by `1` as many times as you need, and use as many dimensions as you need. You can construct a unit of any dimension by multiplying all these `1`’s together. Any other number just tells you how many times you repeat this unit. Like This site uses Akismet to reduce spam. Learn how your comment data is processed.

crawl-data/CC-MAIN-2021-17/segments/1618039554437.90/warc/CC-MAIN-20210421222632-20210422012632-00466.warc.gz

Typically, one wouldn’t think to ask a geologist about the most pressing issues in evolutionary biology. Yet, for some biologists, rock formations and fossil records—which have only gained the attention of natural scientists in the last 50 years—provide a plentiful source of untapped information about the history of life on Earth. A recent study released in the journal Proceedings of the National Academy of Sciences applies geological theory to questions of how some of Earth’s earliest eukaryotes—tiny, multicellular organisms—survived one of the coldest periods of Earth’s history. Using the presence of mineral deposits in rock formations across the globe, Maxwell Lechte, lead author of the study and postdoctoral fellow in McGill’s Department of Earth and Planetary Sciences, and his colleagues proved that liquid water containing a relatively high abundance of oxygen was present underneath the ice masses of what scientists now call ‘Snowball Earth.’ A tumultuous period in Earth’s geological history, Snowball Earth occurred over 700 million years ago during the Cryogenian ice age. During this period, ice was almost everywhere. The proliferation of ice masses across the globe caused the separation of the oceans from the atmosphere, blocking out the sun and plunging earth under ice sheets up to two miles thick. “Reconstructing ice ages using geology is quite a well-established technique because they are quite apparent in the geological record,” Lechte said in an interview with The McGill Tribune. “Ice sheets are extremely powerful erosive forces […] and are quite distinctive in the rock record.” The eternal frigid waste of the Cryogenian Period presents a perplexing question for evolutionary biologists: How complex eukaryotic life, which requires oxygen to live, survived though 100 million years of frozen temperatures when the waters of the ocean contained little to no oxygen. Luckily for baffled biologists, a geologist was asking the same thing. “The question I was interested in is [was] what effect this would have on the biosphere,” Lechte said. “There is no doubt that covering the whole planet in ice would reduce the amount of habitable area.” Scientific consensus has approximated that complex eukaryotic life was first evolving around the same time that Earth was becoming covered in ice. “If you cover the whole oceans in ice, that separates them from the atmosphere, and so all these more complex life forms should have been killed off,” Lechte said. The authors assumed that since eukaryotes survived the ice age, complex life living underneath massive ice shelves must have gotten oxygen from a source other than the atmosphere. “Iron is soluble in sea water under anoxic conditions, so when there is no oxygen around, iron can dissolve,” Lechte said. “But as soon as you have any oxygen around, iron rusts out of the water and becomes insoluble and deposits out as solid iron oxides. The fact we had found all these [iron oxides] in glacial deposits suggests that something interesting was going on with oxygen.” From these observations, Lechte, along with researchers at the University of Melbourne, proposed that meltwater may be the key to supplementing oxygen to organisms living under the ice’s surface. They suggested that air bubbles from the atmosphere had become trapped in the ice shelves as they were forming. These air bubbles, which contained ample oxygen, then became a vital lifeline for marine eukaryotes that could survive off of meltwater that contained much higher levels of oxygen. The implication that oxygen stores could be found on Earth hundreds of millions of years ago is an invaluable link in the evolutionary chain of events. These questions about the long-term evolution of complex life date back as far as Darwin himself. “A lot of our understanding of the past Earth is based on modern Earth,” Lechte said. “People historically have assumed Earth has just always been this way, but we have come to understand these changes as a recent phenomenon.” Layers of iron-rich glacial deposits exposed in the desert illustrate the history of marine environments during Snowball Earth (Death Valley, California). (Maxwell Lechte / The McGill Tribune) A small dropstone within marine sediments with red layers of iron oxides tells the tale of marine oxygenation during Snowball Earth (Ikara-Flinders Ranges, South Australia). (Ashleigh Hood / The McGill Tribune)

<urn:uuid:f7986f4f-26c6-4295-a84a-936008f74952>

Hopi (hōˈpē) [key], group of the Pueblo, formerly called Moki, or Moqui. They speak the Hopi language, which belongs to the Uto-Aztecan branch of the Aztec-Tanoan linguistic stock, at all their pueblos except Hano, where the language belongs to the Tanoan branch of the Aztec-Tanoan linguistic stock (see Native American languages). They occupy several mesa villages in NE Arizona and in 1990 numbered close to 12,000. In 1540, they were visited by some of Francisco Coronado's men under Pedro de Tovar, but because of their geographical isolation they remained more independent of European influence than other Pueblo groups. The Spanish began to establish missions in 1629 at the Hopi pueblos of Awatobi, Oraibi, and Shongopovi. These missions were destroyed in the revolt of 1680 (see Popé), and when the residents of Awatobi invited the missionaries to return, the other Hopi destroyed their village. After the revolt, pueblos in the foothills were abandoned and new villages were built on the mesas for defense against possible attack by the Spanish. The pueblo of Hano was built by the Tewa, who had fled from the area of the Rio Grande valley that the Spanish reconquered. During the 18th and 19th cent., the Hopi were subjected to frequent raids by the neighboring Navajo. The region was pacified by the U.S. army in the late 19th cent., and a Hopi reservation was established in 1882, but the ambiguous status of much of the reservation enabled Navajo populations to encroach on traditional Hopi lands. By the 1960s and 70s, Navajo expansion on lands set aside for joint use provoked court action and led to a partition of the disputed land. Amid bitter conflict, over 10,000 Navajo and fewer than 100 Hopi were relocated from the partitioned lands. A court decision in 1992 assigned most of the land still in dispute to the Navajo. Some Navajo were permitted to remain on Hopi land under 75-year leases. The Hopi are sedentary farmers, mainly dependent on corn, beans, and squash; they also raise wheat, cotton, and tobacco, and herd sheep. Each village is divided into clans and is governed by a chief, who is also the spiritual leader. Political and religious duties revolve around the clans. The Badger clan, for instance, still conducts the kachina (fertility) ceremony, and the Antelope and Snake clans perform the well-known snake dance at Walpi and other pueblos. A Hopi tribal council and constitution were established in 1936, but internal dissension has limited tribal unity. See J. Kammer, The Second Long Walk (1980); S. Rushforth and S. Upham, A Hopi Social History (1992). More on Hopi from Infoplease: See more Encyclopedia articles on: North American indigenous peoples

<urn:uuid:31d2a667-e56d-4268-a2e1-11efb4332c3c>

# Divisible rules for 3 and 9 in a relationship ### Divisibility Rules for 3 and 9 A listing of divisibility rules with illustrated examples and explanations. Know divisibility rules from 2 to 10 with examples. Download FREE divisibility rules pdf chart, flash cards and math worksheets for rules on. In this post we are going to learn about the criteria of divisibility by 3, 4, 9 and We will show some example for each case. How about colorful flash cards to remember these rules? Print the flash cards and cut them to form a card. Or buy our Dr. Math Flash cards to get the whole printed set that covers various topics of math including divisibility rules. Once your child is well learned about all the divisibility rules, you could see how much can they apply it. • Divisibility Rules and Finding Factors • Divisibility Rules • The Divisibility Rules: 3, 6, 9 Here are 2 FUN ways in which you can make your kid practice divisibility rules: Math worksheets Math worksheets and printable are fun and interactive way to practice rules of divisibility. Check out more math worksheets on all different math skills for Kindergarten to Grade 5 here. Math games What better way to learn while you play? It is a fun way for repetition of divisibility rules and bonding with family over the board games. So, hope these divisibility rules for kids will help your child in solving math problems like a pro. Do share this article with your fellow parents for their benefit. ## Divisibility Rules for 2 3 4 5 6 7 8 9 – Printable Charts & Flash Cards The Divisibility Rules help us to determine if a number will divide into another number without actually having to divide. There is a divisibility rule for every number. However, some of the rules are easier to use than others. For the rest, it might just be simpler to actually divide. Here is a look at the rules for 3, 6, and 9. ### Divisibility rules The Rule for 3: A number is divisible by 3 if the sum of the digits is divisible by 3. What does this mean? This means that we need to add up the digits in the number and see of the answer is can be divided by 3 without a remainder. Add up the digits. Determine if 3 divides evenly into the sum of So 3 goes evenly into Use the result to determine if 3 goes into 34, Because 3 divides into 18 evenly, 3 also divides evenly into 34, Is the number prime or composite? We used the procedure listed below. To determine if a number is prime or composite, follow these steps: Find all factors of the number. If the number has only two factors, 1 and itself, then it is prime. If the number has more than two factors, then it is composite. The above procedure works very well for small numbers. Thus we need a better method for determining if a large number is prime or composite. Every number has one and itself as a factor. Thus, if we could find one factor ofother than 1 and itself, we could prove that is composite. One way to find factors of large numbers quickly is to use tests for divisibility. If one whole number is divisible by another number, then the second number is a factor of the first number.

crawl-data/CC-MAIN-2019-51/segments/1575540521378.25/warc/CC-MAIN-20191209173528-20191209201528-00048.warc.gz

# Substitute Numbers for Letters in Complex Calculations (No Division) In this worksheet, students will replace letters with numbers and carry out simple questions using the four number operations. (The student will type the answer as a number and no prioritisation of calculations is required at this stage.) Key stage:  KS 2 Curriculum topic:   Verbal Reasoning Curriculum subtopic:   Maths Codes Difficulty level: ### QUESTION 1 of 10 Make sure that you’ve got your thinking cap on as we are going to be number decoders! In this activity, we are going to swap numbers for letters to solve maths problems. Let’s take a look at this question first: If a = 1, b = 2, c = 3 and d = 4, what is c x d? This is called algebra, which is where numbers are replaced with letters. We need to swap the letters back into numbers to solve this problem. If we put the numbers back in, then c x d becomes 3 x 4. We know this is 12. The values of the letters can be different in each question. Let’s try this question next: If a = 7, b = 4, c = 8 and p = 3, what is a + b + c? Let’s swap the letters back to numbers. This would be written as: 7 + 4 + 8 = 19. Remember, when you are adding lots of numbers, you can choose the order that you add them in. If there is a mixture of addition and subtraction, you must work it out in the order that it is written. Let’s practise this now: If a = 10, b = 4, c = 3 and d = 9, what is b + d - a + c? Let’s swap the letters back to numbers. This would be written as: 4 + 9 - 10 + 3 = Remember, we must answer this one in order. So 4 + 9 = 13… 13 - 10 = 3… 3 + 3 = 6. Now it’s your turn to crack the codes. Good luck number detective! If A = 10, B = 4, C = 3, D = 9, what is B + D - A + C? Can you choose the correct number sentence from the options below? 4 + 9 - 10 + 3 10 + 9 - 4 + 3 4 + 9 - 3 + 10 9 + 4 - 10 - 3 If a = 10, b = 4, c = 3 and d = 9, what is b + d - a + c? Copy and complete the table with each of the steps you would take to solve the problem. Don't forget to use a space between each number, letter or symbol to be marked correctly. Letters Numbers Answer Step 1 b + d 4 + 9 13 Step 2 b + d - a 13 - 10 Step 3 b + d - a + c Letters Numbers Answer Step 1 Step 2 Step 3 If A = 12, B = 5, C = 4 and D = 7, what is B + D - A + C? Can you choose the correct number sentence from the options below? 12 + 5 - 7 + 3 5 + 7 - 12 + 4 5 + 7 - 4 + 12 12 + 4 - 5 + 7 If a = 12, b = 5, c = 4 and d = 7, what is b + d - a + c? Copy and complete the table with each of the steps you would take to solve the problem. Don't forget to use a space between each number, letter or symbol to be marked correctly. Remember to start Step 2 with the answer from Step 1 and Step 3 with the answer from Step 2. Part of the table has been completed here to help you get started. Letters Numbers Answer Step 1 b + d 5 + 7 12 Step 2 b + d - a 12 - Step 3 b + d - a + c Letters Numbers Answer Step 1 Step 2 Step 3 If a = 5, b = 12, c = 7 and d = 8, what is b + d - a + c? Can you choose the correct number sentence from the options below? 12 + 8 - 7 + 5 8 + 7 - 12 + 5 5 + 8 - 7 + 12 12 + 8 - 5 + 7 If A = 5, B = 12, C = 7 and D = 8, what is B + D - A + C? Copy and complete the table with each of the steps you would take to solve the problem. Don't forget to use a space between each number, letter or symbol to be marked correctly. Remember to start Step 2 with the answer from Step 1 and Step 3 with the answer from Step 2. Part of the table has been completed here to help you get started. Letters Numbers Answer Step 1 B + D 12 + 8 Step 2 B + D - A Step 3 B + D - A + C Letters Numbers Answer Step 1 Step 2 Step 3 If a = 9, b = 15, c = 6 and d = 3, what is b + d - a + c? Can you choose the correct number sentence from the options below? 9 + 15 - 6 + 3 6 + 15 - 9 + 3 15 + 3 - 9 + 6 15 + 9 - 3 + 6 If A = 9, B = 15, C = 6 and D = 3, what is B + D - A + C? Copy and complete the table with each of the steps you would take to solve the problem. Don't forget to use a space between each number, letter or symbol to be marked correctly. Remember to start Step 2 with the answer from Step 1 and Step 3 with the answer from Step 2. Part of the table has been completed here to help you get started. Letters Numbers Answer Step 1 B + D Step 2 B + D - A Step 3 B + D - A + C Letters Numbers Answer Step 1 Step 2 Step 3 If A = 3, B = 17, C = 4 and D = 6, what is B + D - A + C? Can you choose the correct number sentence from the options below? 17 + 6 - 4 + 3 17 + 6 - 3 + 4 17 + 4 - 6 + 3 17 + 3 - 4 + 6 If a = 3, b = 17, c = 4 and d = 6, what is b + d - a + c? Copy and complete the table with each of the steps you would take to solve the problem. Don't forget to use a space between each number, letter or symbol to be marked correctly. Remember to start Step 2 with the answer from Step 1 and Step 3 with the answer from Step 2. Part of the table has been completed here to help you get started. Letters Numbers Answer Step 1 b + d Step 2 b + d - c Step 3 b + d - c + a Letters Numbers Answer Step 1 Step 2 Step 3 • Question 1 If A = 10, B = 4, C = 3, D = 9, what is B + D - A + C? Can you choose the correct number sentence from the options below? 4 + 9 - 10 + 3 EDDIE SAYS The question tells us that B = 4, D = 9, A = 10 and C = 3. If we swap the letters for the numbers they represent, we find that B + D - A + C becomes 4 + 9 - 10 + 3. As this question has both addition (+) and subtraction (-) in it, we must do the calculations in order to reach the correct answer. • Question 2 If a = 10, b = 4, c = 3 and d = 9, what is b + d - a + c? Copy and complete the table with each of the steps you would take to solve the problem. Don't forget to use a space between each number, letter or symbol to be marked correctly. Letters Numbers Answer Step 1 b + d 4 + 9 13 Step 2 b + d - a 13 - 10 Step 3 b + d - a + c Letters Numbers Answer Step 1 Step 2 Step 3 EDDIE SAYS Completing all the steps in the correct order leaves you with the answer 6. We will build on this technique in the next few questions so that you can answer all maths codes questions independently using these steps. • Question 3 If A = 12, B = 5, C = 4 and D = 7, what is B + D - A + C? Can you choose the correct number sentence from the options below? 5 + 7 - 12 + 4 EDDIE SAYS The question tells us that B = 5, D = 7, A = 12 and C = 4. If we swap the letters for the numbers they represent, we find that B + D - A + C becomes 5 + 7 - 12 + 4. As this question has both addition (+) and subtraction (-) in it, we must do the calculations in order to reach the correct answer. • Question 4 If a = 12, b = 5, c = 4 and d = 7, what is b + d - a + c? Copy and complete the table with each of the steps you would take to solve the problem. Don't forget to use a space between each number, letter or symbol to be marked correctly. Remember to start Step 2 with the answer from Step 1 and Step 3 with the answer from Step 2. Part of the table has been completed here to help you get started. Letters Numbers Answer Step 1 b + d 5 + 7 12 Step 2 b + d - a 12 - Step 3 b + d - a + c Letters Numbers Answer Step 1 Step 2 Step 3 EDDIE SAYS Completing all the steps in the correct order leaves you with the answer 4. Did you feel more confident completing the grid this time? We will practise using it more in the rest of the questions in this activity. • Question 5 If a = 5, b = 12, c = 7 and d = 8, what is b + d - a + c? Can you choose the correct number sentence from the options below? 12 + 8 - 5 + 7 EDDIE SAYS The question tells us that b = 12, d = 8, a = 5 and c = 7. If we swap the letters for the numbers they represent, we find that b + d - a + c becomes 12 + 8 - 5 + 7. As this question has both addition (+) and subtraction (-) in it, we must do the calculations in order to reach the correct answer. • Question 6 If A = 5, B = 12, C = 7 and D = 8, what is B + D - A + C? Copy and complete the table with each of the steps you would take to solve the problem. Don't forget to use a space between each number, letter or symbol to be marked correctly. Remember to start Step 2 with the answer from Step 1 and Step 3 with the answer from Step 2. Part of the table has been completed here to help you get started. Letters Numbers Answer Step 1 B + D 12 + 8 Step 2 B + D - A Step 3 B + D - A + C Letters Numbers Answer Step 1 Step 2 Step 3 EDDIE SAYS Completing all the steps in the correct order leaves you with the answer 22. Hopefully you are feeling more an more confident using the grid and these steps to help you reach the correct answer. Let's practice some more... • Question 7 If a = 9, b = 15, c = 6 and d = 3, what is b + d - a + c? Can you choose the correct number sentence from the options below? 15 + 3 - 9 + 6 EDDIE SAYS The question tells us that b = 15, d = 3, a = 9 and c = 6. If we swap the letters for the numbers they represent, we find that b + d - a + c becomes 15 + 3 - 9 + 6. As this question has both an addition (+) and subtraction (-) in it, we must do the calculations in order to reach the correct answer. • Question 8 If A = 9, B = 15, C = 6 and D = 3, what is B + D - A + C? Copy and complete the table with each of the steps you would take to solve the problem. Don't forget to use a space between each number, letter or symbol to be marked correctly. Remember to start Step 2 with the answer from Step 1 and Step 3 with the answer from Step 2. Part of the table has been completed here to help you get started. Letters Numbers Answer Step 1 B + D Step 2 B + D - A Step 3 B + D - A + C Letters Numbers Answer Step 1 Step 2 Step 3 EDDIE SAYS Completing all the steps in the correct order leaves you with the answer 15. • Question 9 If A = 3, B = 17, C = 4 and D = 6, what is B + D - A + C? Can you choose the correct number sentence from the options below? 17 + 6 - 3 + 4 EDDIE SAYS The question tells us that B = 17, D = 6, A = 3 and C = 4. If we swap the letters for the numbers they represent, we find that B + D - A + C becomes 17 + 6 - 3 + 4. As this question has both an addition (+) and subtraction (-) in it, we must do the calculations in order to reach the correct answer. • Question 10 If a = 3, b = 17, c = 4 and d = 6, what is b + d - a + c? Copy and complete the table with each of the steps you would take to solve the problem. Don't forget to use a space between each number, letter or symbol to be marked correctly. Remember to start Step 2 with the answer from Step 1 and Step 3 with the answer from Step 2. Part of the table has been completed here to help you get started. Letters Numbers Answer Step 1 b + d Step 2 b + d - c Step 3 b + d - c + a Letters Numbers Answer Step 1 Step 2 Step 3 EDDIE SAYS Completing all the steps in the correct order leaves you with the answer 24. Hopefully you are feeling grid-tastic now, and ready to apply this technique to more maths codes soon! ---- OR ---- Sign up for a £1 trial so you can track and measure your child's progress on this activity. ### What is EdPlace? We're your National Curriculum aligned online education content provider helping each child succeed in English, maths and science from year 1 to GCSE. With an EdPlace account you’ll be able to track and measure progress, helping each child achieve their best. We build confidence and attainment by personalising each child’s learning at a level that suits them. Get started Start your £1 trial today. Subscribe from £10/month. • Tuition Partner

crawl-data/CC-MAIN-2020-05/segments/1579251678287.60/warc/CC-MAIN-20200125161753-20200125190753-00348.warc.gz

U.S. researchers have demonstrated a technology that uses the sun’s heat to convert carbon dioxide and water into the building blocks of traditional fuels, a reverse combustion process that may emerge as a practical alternative to sequestration of CO2 emissions from power plants. The prototype “Sunshine to Petrol” system, developed by Sandia National Laboratories in New Mexico, uses concentrated solar energy to trigger a thermo-chemical reaction in an iron-rich composite located inside a two-sided cylindrical chamber. The iron oxide is designed to lose an oxygen molecule when exposed to 1,500 degree C heat, and then retrieve an oxygen molecule when it is cooled down, essentially converting an incoming supply of CO2 into an outgoing stream of carbon monoxide. Additionally, when researchers pump water into the chamber rather than CO2, the machine produces hydrogen. Combining those retrieved gases — hydrogen and carbon monoxide — they are able to create syngas, which can be used as a fuel. While researchers say the technology likely will not be ready for market for 15 to 20 years, it could one day become a practical way to recycle CO2. “It’s a productive utilization of CO2 that you might capture from a coal plant, a brewery, and similar concentrated sources,” said James Miller, a Sandia chemical engineer. This piece originally appeared on Yale Environment 360. but it still means burning a fuel to power cars, generators etc. I think it is a lot more important to switch to a form of power that does not involve the possible emmission (because we won't be able to convert all the C02) of greenhouse gasses. And besides, once the syngas is formed and used there will just be another round of C02 to recapture. Agree with Joe. Why not use the solar power directly? For energy storage? I'm sure there are other technologies that won't take 15-20 years to mature. File under 'interesting distraction' As with anything we can't expect one "alternative fuel source" to power the world of tomorrow (or today for that matter), but this development does provide a relatively low energy way of generating syngas that can then be converted to synthetic fuels via Fischer Tropsch that can be used via the existing infrastructure to power cars etc; or can be used for generating solely hydrogen (as mentioned) that can power either mobile or static fuel cells (located in factories and feeding electricity into the grid). I wouldn't be so quick to say "File under 'interesting distraction'" as this is a low energy method (most aren't) for generating alternative fuels that can be used with existing infrastructures (electicity grid and fuel stations)

<urn:uuid:bfc46957-700d-473d-b587-455ec7025b42>

## Cross Section Area Of A Wire How can you calculate the surface area of a wire? What is the formula for cross-sectional area? This article will answer these questions and more. Keep reading to learn how to calculate the surface area of a wire. In addition, we’ll explain what the cross-sectional area of a wire is made up of. Once you understand these terms, you’ll be able to calculate the cross-sectional area of a wire. ## Cross section area of a wire The cross section area of a wire is the area of a circular or elliptical plane cut through an object at a right angle to its length. A wire with a circular cross section will have an area equal to the diameter of the wire times its cross section angle. On the other hand, a rectangular block cut at an angle will have an area equal to its cross section angle. Therefore, when you measure the cross section area of a wire, you must first determine how thick the wire is. A wire’s cross section area is calculated by dividing the length of one side by its width. Then, you multiply the two sides by their squares. For example, a wire with a thickness of 3/8 inch and a width of 4 inches is equal to 0.375 inches in cross section area. By converting these measurements, you get a measurement of the wire’s diameter in square inches. Then, you multiply this value by 4,000 to get the square area of the wire. ## What is the formula for cross-sectional area? When measuring the cross-sectional area of a wire, you will need to know its diameter and length. Fortunately, the formula for cross-sectional area is fairly straightforward. A wire’s cross-sectional area is equal to the square of its diameter, measured in mils. It can be calculated by drawing a wire rod. You can also find the cross-sectional area of a wire by squaring its diameter in inches. To calculate the cross-sectional area of a wire, you multiply its length by its diameter. However, the formula assumes a clean cut of 90 degrees. In fact, you can get a larger result if you make a 45-degree cut. The area of a wire in a circle is approximately 1000 mils per square inch. Multi-strand cables have a cross-sectional area that is 4,000 mils squared. To calculate the cross-sectional area of a wire, first determine its average diameter. Divide this number by 10. Next, multiply the result by the number of cables. For example, a section of wire with a diameter of 16 mm needs 32 amperes of current. You can round down to four millimeters for ease of calculations. The formula for cross-sectional area of a wire falls within the tabular data category. ## What is area of cross section in electricity? The area of a conductor’s cross section is measured in mm2. It is equal to the surface area of a circle divided by the radius of the object. In the case of electricity, the area of a wire’s cross section is equal to the surface area of a circle with the same radius. The area of a vein will always be round. When calculating resistance, the diameter of a vein must be larger than the cross section of the conductor. The resistance of a wire is the difficulty of allowing current to flow through it. A long wire has a greater resistance than a short one because electrons collide with more ions as they pass through. A thin wire has fewer spaces for free electrons to pass through. The resistance and area of cross section are inversely related. But you’ll find this formula confusing when you need to consider multiple events. ## What is the surface area of a wire? A wire’s surface area can be measured by calculating its cross-sectional area. The surface area of one strand of 0.20mm diameter copper wire is equivalent to the surface area of a circle of radius r. This measurement can be rounded to the nearest hundredth of a meter. Alternatively, you can find out the surface area of a cylinder made of one mm wire by using a known formula. A large-diameter wire has a higher surface area than a small-diameter wire. Wires with smaller diameters are typically measured in millimeters. In a nutshell, wires can be measured in square units or in circular mils. The cross-sectional area of wire is most easily calculated in circular mils rather than in square units, as the measurement scale of wire size is inverse. ## What shape is the cross-sectional area of a wire? The cross-sectional area of a wire is equal to the area of a circle of diameter d with radius r. As the diameter of the wire is larger than its thickness, the area will always be larger. This measurement is also useful to understand the differences between stranded wire and solid wire. The cross-sectional area of wires can also be used to determine the resistance. To understand the formula, first we need to define the cross-section. A cross-section is the common region of a 3D object. For example, a long cylindrical tube will have a cross-section that is a concentric circle. A beam will be named based on the shape of its cross-section. Basically, the area of a cross-section is the same as its height, width, and thickness. A cross-section calculator will give you the cross-sectional area of a cylinder of diameter 10, height H, and thickness t1. A wire can be circular or oval in cross-section. An ellipsoidal cross-section is also possible. Both wire shapes have their center waist narrower than the rest. The S-shaped wire has a much thinner wall and beam than a Z-shaped wire. This means that the S-shaped wire is easier to bend than a Z-shaped one. In general, the cross-sectional area of a wire will be greater than that of a square. ## Is cross-sectional area the same as diameter? A cross-sectional area is the squared length of one side of a conductor. Then, multiply the square length of the other side. For example, take a rectangular conductor with a thickness of 3/8 inch and a width of four inches. The thickness is expressed as 0.375 inch. This is equivalent to 4,000 mils. Similarly, a width expressed in inches is equal to 375 mils. Then, multiply 375 mils by 4,000 mils to find the cross-sectional area. A wire’s cross-sectional area is measured using the formula: A = 1/pd2, where p is the length in feet. Its diameter is the area of a circle with radius r. The cross-sectional area of an n-gauge wire is equal to the square of its diameter. Once you have found the cross-sectional area of the wire, you can calculate the average diameter of the wire. ## What is the difference between area and cross-section. The size of a wire can be measured using the area and cross section of the conductor. The area of a wire refers to the space in which the copper wires can pass. It is important to note that the cross-section area and diameter are not the same. Likewise, stranded wire has a larger cross-section than solid wire. So, the size of a solid wire is more important than its cross-sectional area. The cross-sectional area of a wire is smaller than its overall surface area. Generally speaking, large-diameter wires have greater cross-sectional area than small-diameter wires. The cross-sectional area of a wire can be expressed in either square units or in circular mils. The area of a wire can also be expressed in the gauge scale. The circular-mil measurement of a wire is more convenient to calculate, as it eliminates the “pi” and d/2 (radius) factors. The cross-sectional area of a wire affects its resistance. A wider wire has lower resistance than a thinner one. Therefore, the wider the cross-section area, the lower the resistance of the wire. To further understand this, consider the example of a water pipe. The wider a pipe is, the more water it flows. Therefore, a wire with a wider cross-section area has less resistance to the flow of electric charge. ## What is area formula? What is the area formula for cross section of a wire, and how do I find its area? Wire cross sections are shaped like a circle, but the surface area of each section varies. Wires are a mix of different materials, and one type is generally more dense than another. One type of wire is stranded, which is a single-core wire that has been twisted together. The cross section of a wire is a two-dimensional representation of the object. When cutting a solid wire into multiple sections, the two-dimensional slices of the wire will be different. The cross-section area is known as the SS, and it is measured in mm2. A stranded wire will have a greater area than a solid wire. Both types of wire have a different resistance. One method of calculating the cross-section area of a wire is to measure the diameter of the wire. This measurement is easy. Take a long piece of wire and wind it around a pencil until the “tails” fit tight together. Make sure to use full turns that fit tightly and have no gaps between them. You’ll need to divide the length of the segment by the number of turns to determine the diameter. For example, if the wire has 11 turns, the resulting diameter will be 7.5 mm. Then divide that number by 11 and get 0.68 mm. ## How to Find the Cross-Sectional Area of a Rectangle Duct A rectangular duct can be divided into sixteen sections. Each section indicates the average velocity of the air flowing through the duct. The cross-sectional area of a duct can be calculated using a formula that represents the cross-sectional area and velocity of the air. The cross-sectional area of a rectangular duct is equal to the flow rate of one cubic meter of air per second, where V is the velocity of the air. The area of a solid depends on its shape and the angle between its axis of symmetry and the plane in which it intersects. The area of a rectangular solid is equal to the base area x its height. The cross-section area of a rectangular duct is equal to the base area plus its height. If you want to calculate the cross-sectional area of a square, you would multiply the width of the cylinder by the height. A duct’s cross-sectional area is measured in square inches, and this value can be calculated from the length and circumference of the cylinder. The resulting square area is then multiplied by the radius of the cylinder. If the duct is round, the area of the cylinder is equal to p*R2. This is true for both rectangular and oval shaped ducts, but rectangular ducts are more accurate. ## What is Meant by Cross-Sectional Area of Conductor? The cross-sectional area of a conductor is the surface area that is the same lengthwise, no matter the configuration. Its area can be measured in square mils or in the actual cross-section of the conductor. The square mil is a unit of measurement, where one mil equals the area of a square with sides of 1 mil. A 3/8-inch conductor, for example, has a cross-sectional area of 3/8 of an inch, and it is 4 inches wide. Therefore, a 3/8-inch conductor is 4 inches wide and 3/8 inch thick. The area of a circular conductor is equal to 0.375 inch. A rectangular conductor will have an area of 9 square mils, so a 3/8-inch square will have a cross-sectional area of A cable is a small pipe. The configuration determines its outline. For example, if you cut a round metal rod in half, the cross-section will be two circles of a specific thickness. The cross-sectional area of a conductor (SS) is measured in mm2, whereas the area of a vein is round. Using this formula, you can find the cross-sectional area of a conductor by multiplying the radius of the vein by its circumference, R. Another application for cross-sectional area is in nuclear physics. The effective size of a nuclear atom is defined by the cross-section of the nucleus. The cross-sectional area of a nuclear atom is the area of a circle divided parallel to the base, and the probability of the neutron interacting with the target atom is expressed by its cross-section. Nuclear fission relies on this mechanism.

crawl-data/CC-MAIN-2022-49/segments/1669446710869.86/warc/CC-MAIN-20221201185801-20221201215801-00458.warc.gz

#### Need Help? Get in touch with us # Absolute Value Function Sep 16, 2022 ## Key Concepts • Graph the absolute value function • Transform the absolute value function • Interpret the graph of a function • Determine rate of change ### Graph the Absolute Value Function What are the features of the graph of f(x) = |x|? Make a table of values and graph the absolute value function f(x) = |x|. The graph has an axis of symmetry, which intersects the vertex and divides the graph into two sections, or pieces, that are images of each other under a reflection. ### Transform the Absolute Value Function A. How do domain and range of g(x) = 2 |x| compare with the domain and range of f(x) = |x|? Compare the graphs of g and f. The domain of g and f are all real numbers. Because the absolute value expression produces only non-negative values, the range of f is y0. Multiplying IxI by a positive factor, in this case, 2, yields non-negative outputs, so the range of g is also y0 The domain and range of the function g are the same as those of function f. B. How do domain and range of h(x) = –1|x| compare with the domain and range of f(x) = |x|? Compare the graphs of h and f. The domain of h and the domain of f is all real numbers. The range of f is:  y0. Multiplying IxI by a negative factor, -1, yields non-positive outputs, so the range of h is also  y0. The domain of h is the same as the domain of f. The range of h is the opposite of the range of f. ### Interpret the Graph of a Function Jay rides in a boat from Monroe County to Collier County. The graph of the function d(t) = 30 |t-1.5| shows the distance of the boat in miles from the county line at t hours. Assume, the graph shows Jay’s entire trip. 1. How far does Jay travel to visit his friend? Jay began his trip 45 mi from the county line, traveled towards the county line, which he crossed after an hour and a half. He then traveled away from the county line and was 45 mi from the county line after 3 h. He traveled a total of 90 mi. 1. How does the graph relate to the domain and range of the function? Since Jay’s entire trip is 3 h, the domain of the function is:  0t 3 For 1.5 < t < 3, the section of the domain 0 < t < 1.5, his distance to the border is decreasing. For, his distance from the border is increasing. The maximum and minimum values on the graph are 45 and 0, so the range of the function is   0d(t) 45. ### Determine Rate of Change The graph shows Jay’s boat ride across the state line from the previous example. What is the rate of change over the interval  2 ≤ t ≤ 2.5 What does it mean in terms of the situation? Use the slope formula to determine the rate of change from t = 2 to t = 2.5. Use the points from (2, 15) and (2.5, 30). The rate of change over this interval is 30. The rate of change represents the speed of the boat in miles per hour. Since the rate of change is positive, Jay’s distance from the border is increasing. The boat is traveling at 30 mi/h away from the border. ### Questions Question 1 What are the domain and range of f(x) = |x|? Solution: Below is the graph of f(x) = |x|. As we can see in the graph, the value of x can be anything. So, the domain of f  is all real numbers. In the graph, the value of f(x) is always positive or zero. So, the range of the function is   y0. Question 2 A cyclist competing in a race ride past a water station. The graph of the function d(t) = (1/3) |t – 60| shows her distance from the water station at t minutes. Assume, the graph represents the entire race. What does the graph tell you about her race? Solution: The cyclist traveled 20 km towards the water station in one hour and at the end of one hour, she reached the water station. Then she started traveling away from the water station and covered the distance of 20 km away from the water station in the next one hour. So, the cyclist finished her entire race of 40 km in two hours. Question 3 Kata gets on a moving walkway to the airport. The, 8 s after she gets on, she taps Lisa, who is standing alongside the walkway. The graph shows Kata’s distance from Lisa over time. Calculate the rate of change in her distance from Lisa from 6 s to 8 s, and then from 8 s to 12 s. What do the rates of change mean in terms of Kata’s movement? Solution: Let’s find the value of d(t) at t = 6 s, t = 8 s and t = 12 s. t = 6, d(t) = 4 t = 8, d(t) = 0 t = 12, d(t) = 10 Rate of change from 6 s to 8 s: Use the points (6, 4) and (8, 0). m = (y2 – y1)/ (x2 – x1) = (0 – 4)/ (8 – 6) = (-4)/2 = -2 ft/s Rate of change from 8 s to 12 s: Use the points (8, 0) and (12, 10). m = (y2 – y1)/ (x2 – x1) = (10 – 0)/ (12 – 8) = 10/4 = 2.5 ft/s Since the rate of change from 6 s to 8 s is negative, Kata’s distance from Lisa is decreasing and she is traveling at the speed of 2 ft/s toward Lisa. And then from 8 s to 12 s, the rate of change is positive. So, Kata’s distance from Lisa is increasing, and is traveling at the speed of 2.5 ft/s moving away from Lisa. ## Exercise Graph the following functions: 1. f(x) = 0.6|x| 2. f(x) = (1/7) |x| 3. f(x) = |x – 100| 4. f(x) = 3|x| 5. f(x) = 2Ix – 100I 6. f(x) = (1/3) |x – 100| 7. f(x) = 4.5|x| 8. f(x) = 0.5|x + 10| 9. f(x) = 10|x| 10. f(x) = 9|x + 100| #### Addition and Multiplication Using Counters & Bar-Diagrams Introduction: We can find the solution to the word problem by solving it. Here, in this topic, we can use 3 methods to find the solution. 1. Add using counters 2. Use factors to get the product 3. Write equations to find the unknown. Addition Equation: 8+8+8 =? Multiplication equation: 3×8=? Example 1: Andrew has […] #### Dilation: Definitions, Characteristics, and Similarities Understanding Dilation A dilation is a transformation that produces an image that is of the same shape and different sizes. Dilation that creates a larger image is called enlargement. Describing Dilation Dilation of Scale Factor 2 The following figure undergoes a dilation with a scale factor of 2 giving an image A’ (2, 4), B’ […] #### How to Write and Interpret Numerical Expressions? Write numerical expressions What is the Meaning of Numerical Expression? A numerical expression is a combination of numbers and integers using basic operations such as addition, subtraction, multiplication, or division. The word PEMDAS stands for: P → Parentheses E → Exponents M → Multiplication D → Division  A → Addition S → Subtraction         Some examples […]

crawl-data/CC-MAIN-2024-33/segments/1722640723918.41/warc/CC-MAIN-20240808062406-20240808092406-00574.warc.gz

# Eureka Math Algebra 2 Module 1 Lesson 39 Answer Key ## Engage NY Eureka Math Algebra 2 Module 1 Lesson 39 Answer Key ### Eureka Math Algebra 2 Module 1 Lesson 39 Opening Exercise Answer Key Rewrite each expression as a polynomial in standard form. a. (x + i)(x – i) (x + i)(x – i) = x2 + ix – ix – i2 = x2 – i2 = x2 – (-1) = x2 + 1 b. (x + 5i)(x – 5i) (x + 5i)(x – 5i) =x2 + 5ix – 5ix – 25i2 = x2 – 25i2 = x2 – 25(- 1) =x2 + 25 c. (x – (2 + i))(x – (2 – i)) (x – (2 + i))(x – (2 – 1)) = x2 – (2 + i)x – (2 – i)x + [(2 + i)(2 – t)] = x2 – 2x – ix – 2x + ix + [4 – i2] = x2 – 4x + [4-(-1)] = x2 – 4x + 5 ### Eureka Math Algebra 2 Module 1 Lesson 39 Exercise Answer Key Factor the following polynomial expression into products of linear terms. Exercise 1. x2 + 9 x2 + 9 = (x + 3i) (x – 3i) Exercise 2. x2 + 5 x2 + 5 = (x + i√5) (x – i√5) Exercise 3. Consider the polynomial P(x) = x4 – 3x2 – 4. a. What are the solutions to x4 – 3x2 – 4 = 0? x4 – 3x2 – 4 = 0 (x2)2 – 3x2 – 4 = 0 (x2 + 1)(x2 -4) = 0 (x + i) (x – i) (x + 2) (x – 2) = 0 The solutions are -i, i, -2, and 2. b. How many x-intercepts does the graph of the equation y = x4 – 3x2 – 4 have? What are the coordinates of the x-intercepts? The graph of y = x4 – 3x2 – 4 has two x-intercepts: (-2, 0) and (2, 0). c. Are solutions to the polynomial equation P(x) = 0 the same as the x-lntercepts of the graph of y = Justify your reasoning. No. Only the real solutions to the equation are x-intercepts of the graph. By comparing the graph of the polynomial in part (b) to the equation’s solutions from part (c), you can see that only the real number solutions to the equation correspond to the x-intercepts in the Cartesian plane. Exercise 4. Write a polynomial P with the lowest possible degree that has the given solutions. Explain how you generated each answer. a. – 2, 3, – 4i, 4i The polynomial P has two real zeros and two complex zeros. Since the two complex zeros are members of a conjugate pair, P may have as few as four total factors. Therefore, P has degree at least 4. P(x) = (x + 2) (x – 3) (x + 4i) (x – 4i) = (x2 – x – 6) ( x2 – 16i2) = (x2 – x – 6) (x2 + 16) = x4 – x3 – 6x2 + 16x2– 1 6x – 96 = x4 – x3 + 10x2 – 16x -96 b. – 1, 3i The polynomial P has one real zero and two complex zeros because complex zeros come in pairs. Since 3 i and – 3i form a conjugate pair, P has at least three total factors. Therefore, P has degree at least 3. P(x) = (x + 1) (x – 3i) (x + 3i) = (x + 1) (x2 – 9i2) = (x + 1) (x2 + 9) = x3 + x2 + 9x + 9 c. 0, 2, 1+i, 1-i Since 1 + i and 1 – i are complex conjugates, P is at least a 4th degree polynomial. P(x) = x(x – 2) (x – (1 + i)) (x – (1 – i)) = x(x – 2) [(x – 1) -i [(x – 1) + i] = x(x -2) [(x – 1)2 – i2] = x(x – 2) [(x2 – 2x + 1) + 1] = x(x – 2) (x2 – 2x + 2) = x(x3 – 2x2 + 2x – 2x2 + 4x – 4) = x (x3 – 4x2 + 6x – 4) = x4 – 4x3 + 6x2 – 4x d. √2, -√2, 3, 1 + 2i Since 1 + 2i is a complex solution to P(x) = 0, its conjugate, 1 – 2i, must also be a complex solution. Thus, P is at least a fifth-degree polynomial. P(x) = (x – √2) (x + √2) (x – 3) (x – (1 + 2i)) (x – (1 – 2i)) = (x2 – 2) (x – 3) [(x – 1) – 2i] [(x – 1) + 2i] = (x2 – 2) (x – 3) [(x – 1)2 – 4i2] = (x2 – 2) (x – 3) [(x2 – 2x + 1) + 4] = (x2 – 2) (x – 3) (x2 – 2x + 5) = (x3 – 3x2 – 2x + 6) (x2 – 2x + 5) = x5 – 5x4 + 9x3 – 5x2 – 22x + 30 e. 2i, 3 – i The complex conjugates of 2i and 3 – i are -2i and 3 + i, respectively. So, P is at least a fourth-degree polynomial. P(x) = (x – 2i) (x + 2i)( x – (3 – i))(x – (3 + i)) = (x2 – 4i2) [(x – 3) + i] [(x – 3) – i] = (x2 + 4) [(x – 3)2 – i2] = (x2 + 4) [(x2 – 6x + 9) + 1] = (x2 + 4) (x2 – 6x + 10) = x4 – 6x3 + 14x2 – 24x + 40 ### Eureka Math Algebra 2 Module 1 Lesson 39 Problem Set Answer Key Question 1. Rewrite each expression in standard form. a. (x + 3i) (x – 3i) x2 + 32 = x2 + 9 b. (x – a + bi) (x – (a + bi)) (x – a + bi) (x – (a + bi)) = ((x – a) + bi) ((x – a) – bi) = (x – a)2 + b2 = x2 – 2ax + a2 + b2 c. (x + 2i)(x – i)(x + i)(x – 2i) (x + 2i) (x – 2i) (x + i) (x – i) = (x2 + 22)( x2 + 12) = (x2 + 4) (x2 + 1) = x4 + 5x2 + 4 d. (x + i)2 ∙ (x – i)2 (x + i)(x – i) ∙ (x + i) (x – i) = (x2 + 1) (x2 + 1) = x4 + 2x2 + 1 Question 2. Suppose in Problem 1 that you had no access to paper, writing utensils, or technology. How do you know that the expressions in parts (a)-(d) are polynomials with real coefficients? In part (a), the identity (x + ai) (x – ai) = x2 + a2 can be applied. Since the number a is real, the resulting polynomial will have real coefficients. The remaining three expressions can all be rearranged to take advantage of the conjugate pairs identity. In parts (c) and (d), regrouping terms will produce products of polynomial expressions with real coefficients, which will again have real coefficients. Question 3. Write a polynomial equation of degree 4 in standard form that has the solutions i, – i, 1, – 1. The first step is writing the equation in factored form: (x + 1) (x – i) (x + 1) (x – 1) = 0. Then, use the commutative property to rearrange terms and apply the difference of squares formula: (x + i) (x – i) (x + 1) (x – 1) = (x2 + 1) (x2 – 1) = x4 – 1. So, the standard form of the equation ¡s x4 – 1 = 0. Question 4. Explain the difference between x-intercepts and solutions to an equation. Give an example of a polynomial with real coefficients that has twice as many solutions as x-intercepts. Write it in standard form. The x-intercepts are the real solutions to a polynomial equation with real coefficients. The solutions to an equation can be real or not reaL The previous problem is an example of a polynomial with twice as many solutions than x intercepts. Or, we could consider the equation x4 – 6x3 + 13x2 – 12x + 4 = 0, which has zeros of multiplicity 2 at both 1 and 2. Question 5. Find the solutions to x4 – 5x2 – 36 = 0 and the x-intercepts of the graph of y = x4 – 5x2 – 36. (x2 + 4) (x2 – 9) = 0 (x + 2i) (x – 2i) (x + 3) (x – 3)= 0 Since the solutions are 2i, -21, 3, and -3, and only real solutions to the equation are x-intercepts of the graph, the x-intercepts are 3 and – 3. Question 6. Find the solutions to 2x4 – 24x2 + 40 = 0 and the x-intercepts of the graph of y = 2x4 – 24x2 + 40. 2(x4 – 12x2 + 20) = 0 2(x2 – 10) (x2 – 2) = 0 Since all of the solutions √10, – √10, √2, and -√2 are real numbers, the x-intercepts of the graph are √10, -√10, √2, and -√2. Question 7. Find the solutions to x4 – 64 = 0 and the x-intercepts of the graph of y = x4 – 64. (x2 + 8) (x2 – 8) = 0 (x + √8i) (x – √8i) (x + √8) (x – √8) = 0 The x-intercepts are 2√2 and – 2√2. Question 8. Use the fact that x4 + 64 = (x2 – 4x + 8) (x2 + 4x + 8) to explain how you know that the graph of y = x4 + 64 has no x-intercepts. You need not find the solutions. The x-intercepts of y = x4 + 64 are solutions to (x2 – 4x + 8) (x2 + 4x + 8) = 0. Both x2 – 4x + 8 = 0 and x2 + 4x + 8 = 0 have negative discriminant values of -16, so the equations x2 – 4x + 8 = 0 and x2 + 4x + 8 = 0 have no real solutions. Thus, the equation x4 + 64 = 0 has no real solutions, and the graph of y = x4 + 64 has no x-intercepts. Since x4 + 64 = 0 has no real solutions, the graph of y = x4 + 64 has no x-intercepts. ### Eureka Math Algebra 2 Module 1 Lesson 39 Exit Ticket Answer Key Question 1. Solve the quadratic equation x2 + 9 = 0. What are the x-lntercepts of the graph of the function f(x) = x2 + 9? x2 + 9 = 0 x = √-9 or x = -√-9 x = 3√-1 or x = – 3√-1 x = 3i or x = – 3i The x-intercepts of the graph of the function f(x) = x2 + 9 are the real solutions to the equation x2 + 9 = 0. However, since both solutions to x2 + 9 = 0 are not real, the function f(x) = x2 + 9 does not have any x intercepts. Question 2. Find the solutions to 2x5 – 5x3 – 3x = 0. What are the x-intercepts of the graph of the function f(x) = 2x5 – 5x3 – 3x? x(x + √3) (x – √3) (x + $$\frac{i \sqrt{2}}{2}$$) (x – $$\frac{i \sqrt{2}}{2}$$) = 0 Thus, x = 0, x = -√3, x = √3, x = –$$\frac{i \sqrt{2}}{2}$$, or x = $$\frac{i \sqrt{2}}{2}$$ The solution are, 0, √3, -√3, $$\frac{i \sqrt{2}}{2}$$ and –$$\frac{i \sqrt{2}}{2}$$

crawl-data/CC-MAIN-2024-18/segments/1712296816942.33/warc/CC-MAIN-20240415045222-20240415075222-00261.warc.gz

MLA Descriptive Essay: Basic Components Of A Good Paper A descriptive essay asks the writer to describe something. Why would you need to describe something? Perhaps your instructor wants to evaluate your writing skills or observational skills. Descriptive writing has useful applications in the real world, such as writing advertisements or product descriptions for catalogs. Regardless of what applications this kind of writing has, you’ve been tasked to write a descriptive essay, and you want to make sure it’s a good one. Here are the basic components you need to write a good descriptive paper. - Focus on the thing to be described. This can be a person, place, or object. It can also be emotional like a memory or an experience you may have had. Flesh out the details of whatever it is you must describe. Brainstorm about every aspect of whatever it is you are describing. Make sure you use sensory details to immerse really the reader. - When you come to describe something, there should be a reason for it. If you describe a person maybe you want the reader to know just how much you admire, or, despise that person. If you are writing about a place or experience, you should let the reader know what that means to you and describe it in such a way that it is evident. - Structure your paper in a way that is easily read. Start with what or who you want to describe. Let the reader know right from the start what your paper is going to be about. Next, let the reader know your reason for writing a descriptive piece about this particular person or thing. Without a reason as to why he/she/it is being described, the reader may get lost in your writing or lose interest. Lastly, highlight the qualities of the person, place, or thing you are describing that most appeal to you, and are the reason that you wrote on this subject in the first place. Use an outline to keep all these ideas organized in the proper order so that you can write your paper more smoothly. Revise your essay and make sure that what you have chosen to describe can be conjured easily in the reader’s mind. If you can’t see, hear, taste, feel, or smell whatever it is you have described, then go back and write it so you can. Otherwise, you would have failed at writing a winning description. If you aren’t fully immersed in the description, then your reader won’t be either.

<urn:uuid:ae61931e-4d01-43b8-9902-ffd96271bbe3>

How do we find polygenic genes? You’ve learnt about the genetic architecture of type 1 and type 2 diabetes, but how have we made these discoveries and how do we go about finding new genes involved? Before 2007 the process was very much a ‘fishing expedition’, scientists had to predict which genes might be involved based on their knowledge of the causes of diabetes and then test those genes in a series of patients. Around 2007 there was a substantial improvement in genomic technology that allowed scientists to test genetic variants across the whole genome in a single experiment at a reasonable cost. Such tests now cost approximately $50 per patient. The method makes use of the phenomenon of linkage disequilibrium, which means that genetic variants that are physically close to each other and not separated by recombination during meiosis will be inherited together. So for example, if variants are on different chromosomes they can only be passed on together by chance, there is a 50% chance that those two variants will appear together in the same sperm or egg cell (gamete). If variants are within a few bases of each other they will almost always be inherited together, so a 100% chance of them being in a gamete together. The chance that they will be together diminishes as the distance between them increases. This means that by testing one genetic variant it gives you information about the genetic variation nearby. Thus by testing variants spread across the genome, you obtain information about variation over a large proportion of the genome. This principle is exploited in Genome-Wide Association Studies (GWAS). Typically around 500,000 variants are directly tested which capture information about several million untested variants, spanning the 20,000 genes in the genome. The variants tested are Single Nucleotide Polymorphisms (SNPs) which are simple variants that differ at a single DNA nucleotide position with generally two alternative bases or alleles and are the most abundant variants in the genome. In GWAS for Type 1 diabetes and Type 2 diabetes, case control (discontinuous phenotype) studies are used and for each SNP the frequency of each allele of a SNP is compared between cases and controls. For continuous phenotypes, like fasting glucose measures, the mean level for each genotype category is compared. Statistical tests are used to determine whether the difference between cases and controls, or the difference in mean for each genotype, is greater than would be expected by chance. A similar test is applied to all the SNPs across the genome. If the strength of the statistical association with having diabetes passes a threshold, we can be confident that a new genetic region for the disease has been found. GWAS have found around 90 genetic regions that are associated with Type 2 diabetes. By identifying genetic regions associated with traits and diseases we can understand more about the biology of those traits, perhaps develop novel therapeutics and start to be able predict individuals at increased risk of developing a disease. © University of Exeter

<urn:uuid:d80722d4-ecda-441c-ae48-1e507c70149e>