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Assume all gases are perfect unless stated otherwise. Note that 1 atm = 1.013 25 bar. Unless otherwise stated, thermochemical data are for 298.15 K. A sample consisting of $2.0 \mathrm{~mol} \mathrm{CO}_2$ occupies a fixed volume of $15.0 \mathrm{dm}^3$ at $300 \mathrm{~K}$. When it is supplied with $2.35 \mathrm{~kJ}$ of energy as heat its temperature increases to $341 \mathrm{~K}$. Assume that $\mathrm{CO}_2$ is described by the van der Waals equation of state, and calculate $\Delta U$. | +2.35 | +2.35 | $\text{kJ}$ | atkins | p2.3(b) | scibench | [
{
"content": "<|im_start|>system\nA conversation between User and Assistant. The user asks a question, and the assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and<answer> </answer> tags, respectively, i.e., <think> reasoning process here </think><answer> answer here </answer>.\n<|im_end|>\n<|im_start|>user\nAssume all gases are perfect unless stated otherwise. Note that 1 atm = 1.013 25 bar. Unless otherwise stated, thermochemical data are for 298.15 K. A sample consisting of $2.0 \\mathrm{~mol} \\mathrm{CO}_2$ occupies a fixed volume of $15.0 \\mathrm{dm}^3$ at $300 \\mathrm{~K}$. When it is supplied with $2.35 \\mathrm{~kJ}$ of energy as heat its temperature increases to $341 \\mathrm{~K}$. Assume that $\\mathrm{CO}_2$ is described by the van der Waals equation of state, and calculate $\\Delta U$.\n<|im_end|>\n<|im_start|>assistant\n<think>",
"role": "user"
}
]
| logic | {
"ground_truth": {
"answer_number": "+2.35",
"unit": "$\\text{kJ}$"
},
"style": "rule"
} | {
"index": 300,
"split": "train"
} |
||
Assume all gases are perfect unless stated otherwise. Unless otherwise stated, thermodynamic data are for 298.15 K. The standard enthalpy of formation of ethylbenzene is $-12.5 \mathrm{~kJ} \mathrm{~mol}^{-1}$. Calculate its standard enthalpy of combustion. | -4564.7 | -4564.7 | $\mathrm{kJ} \mathrm{mol}^{-1}$ | atkins | e2.18(a) | scibench | [
{
"content": "<|im_start|>system\nA conversation between User and Assistant. The user asks a question, and the assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and<answer> </answer> tags, respectively, i.e., <think> reasoning process here </think><answer> answer here </answer>.\n<|im_end|>\n<|im_start|>user\nAssume all gases are perfect unless stated otherwise. Unless otherwise stated, thermodynamic data are for 298.15 K. The standard enthalpy of formation of ethylbenzene is $-12.5 \\mathrm{~kJ} \\mathrm{~mol}^{-1}$. Calculate its standard enthalpy of combustion.\n<|im_end|>\n<|im_start|>assistant\n<think>",
"role": "user"
}
]
| logic | {
"ground_truth": {
"answer_number": "-4564.7",
"unit": " $\\mathrm{kJ} \\mathrm{mol}^{-1}$"
},
"style": "rule"
} | {
"index": 301,
"split": "train"
} |
||
Assume all gases are perfect unless stated otherwise. Unless otherwise stated, thermodynamic data are for 298.15 K. The standard enthalpy of decomposition of the yellow complex $\mathrm{H}_3 \mathrm{NSO}_2$ into $\mathrm{NH}_3$ and $\mathrm{SO}_2$ is $+40 \mathrm{~kJ} \mathrm{~mol}^{-1}$. Calculate the standard enthalpy of formation of $\mathrm{H}_3 \mathrm{NSO}_2$. | $-383$ | -383 | $\mathrm{kJ} \mathrm{~mol}^{-1}$ | atkins | e2.22(a) | scibench | [
{
"content": "<|im_start|>system\nA conversation between User and Assistant. The user asks a question, and the assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and<answer> </answer> tags, respectively, i.e., <think> reasoning process here </think><answer> answer here </answer>.\n<|im_end|>\n<|im_start|>user\nAssume all gases are perfect unless stated otherwise. Unless otherwise stated, thermodynamic data are for 298.15 K. The standard enthalpy of decomposition of the yellow complex $\\mathrm{H}_3 \\mathrm{NSO}_2$ into $\\mathrm{NH}_3$ and $\\mathrm{SO}_2$ is $+40 \\mathrm{~kJ} \\mathrm{~mol}^{-1}$. Calculate the standard enthalpy of formation of $\\mathrm{H}_3 \\mathrm{NSO}_2$.\n<|im_end|>\n<|im_start|>assistant\n<think>",
"role": "user"
}
]
| logic | {
"ground_truth": {
"answer_number": "-383",
"unit": "$\\mathrm{kJ} \\mathrm{~mol}^{-1}$ "
},
"style": "rule"
} | {
"index": 302,
"split": "train"
} |
||
Assume all gases are perfect unless stated otherwise. Unless otherwise stated, thermodynamic data are for 298.15 K. A sample of carbon dioxide of mass $2.45 \mathrm{~g}$ at $27.0^{\circ} \mathrm{C}$ is allowed to expand reversibly and adiabatically from $500 \mathrm{~cm}^3$ to $3.00 \mathrm{dm}^3$. What is the work done by the gas? | -194 | -194 | $\mathrm{~J}$ | atkins | e2.10(a) | scibench | [
{
"content": "<|im_start|>system\nA conversation between User and Assistant. The user asks a question, and the assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and<answer> </answer> tags, respectively, i.e., <think> reasoning process here </think><answer> answer here </answer>.\n<|im_end|>\n<|im_start|>user\nAssume all gases are perfect unless stated otherwise. Unless otherwise stated, thermodynamic data are for 298.15 K. A sample of carbon dioxide of mass $2.45 \\mathrm{~g}$ at $27.0^{\\circ} \\mathrm{C}$ is allowed to expand reversibly and adiabatically from $500 \\mathrm{~cm}^3$ to $3.00 \\mathrm{dm}^3$. What is the work done by the gas?\n<|im_end|>\n<|im_start|>assistant\n<think>",
"role": "user"
}
]
| logic | {
"ground_truth": {
"answer_number": " -194",
"unit": "$\\mathrm{~J}$"
},
"style": "rule"
} | {
"index": 303,
"split": "train"
} |
||
Assume all gases are perfect unless stated otherwise. Unless otherwise stated, thermodynamic data are for 298.15 K. Calculate the work needed for a $65 \mathrm{~kg}$ person to climb through $4.0 \mathrm{~m}$ on the surface of the Earth. | $2.6 \times 10^3$ | 2600 | $\mathrm{J}$ | atkins | e2.1(a)(a) | scibench | [
{
"content": "<|im_start|>system\nA conversation between User and Assistant. The user asks a question, and the assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and<answer> </answer> tags, respectively, i.e., <think> reasoning process here </think><answer> answer here </answer>.\n<|im_end|>\n<|im_start|>user\nAssume all gases are perfect unless stated otherwise. Unless otherwise stated, thermodynamic data are for 298.15 K. Calculate the work needed for a $65 \\mathrm{~kg}$ person to climb through $4.0 \\mathrm{~m}$ on the surface of the Earth.\n<|im_end|>\n<|im_start|>assistant\n<think>",
"role": "user"
}
]
| logic | {
"ground_truth": {
"answer_number": "2600",
"unit": "$\\mathrm{J}$"
},
"style": "rule"
} | {
"index": 304,
"split": "train"
} |
||
What pressure would $131 \mathrm{g}$ of xenon gas in a vessel of volume $1.0 \mathrm{dm}^3$ exert at $25^{\circ} \mathrm{C}$ if it behaved as a van der Waals gas? | 22 | 22 | $\mathrm{atm}$ | atkins | e1.1(a)(b) | scibench | [
{
"content": "<|im_start|>system\nA conversation between User and Assistant. The user asks a question, and the assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and<answer> </answer> tags, respectively, i.e., <think> reasoning process here </think><answer> answer here </answer>.\n<|im_end|>\n<|im_start|>user\nWhat pressure would $131 \\mathrm{g}$ of xenon gas in a vessel of volume $1.0 \\mathrm{dm}^3$ exert at $25^{\\circ} \\mathrm{C}$ if it behaved as a van der Waals gas?\n<|im_end|>\n<|im_start|>assistant\n<think>",
"role": "user"
}
]
| logic | {
"ground_truth": {
"answer_number": "22",
"unit": "$\\mathrm{atm}$ "
},
"style": "rule"
} | {
"index": 305,
"split": "train"
} |
||
A constant-volume perfect gas thermometer indicates a pressure of $6.69 \mathrm{kPa}$ at the triple point temperature of water (273.16 K). What change of pressure indicates a change of $1.00 \mathrm{~K}$ at the latter temperature? | 0.0245 | 0.0245 | $\mathrm{kPa}$ | atkins | p1.5(c) | scibench | [
{
"content": "<|im_start|>system\nA conversation between User and Assistant. The user asks a question, and the assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and<answer> </answer> tags, respectively, i.e., <think> reasoning process here </think><answer> answer here </answer>.\n<|im_end|>\n<|im_start|>user\nA constant-volume perfect gas thermometer indicates a pressure of $6.69 \\mathrm{kPa}$ at the triple point temperature of water (273.16 K). What change of pressure indicates a change of $1.00 \\mathrm{~K}$ at the latter temperature?\n<|im_end|>\n<|im_start|>assistant\n<think>",
"role": "user"
}
]
| logic | {
"ground_truth": {
"answer_number": "0.0245",
"unit": "$\\mathrm{kPa}$ "
},
"style": "rule"
} | {
"index": 306,
"split": "train"
} |
||
Assume all gases are perfect unless stated otherwise. Unless otherwise stated, thermodynamic data are for 298.15 K. Calculate the work needed for a $65 \mathrm{~kg}$ person to climb through $4.0 \mathrm{~m}$ on the surface of the Moon $\left(g=1.60 \mathrm{~m} \mathrm{~s}^{-2}\right)$. | $4.2 \times 10^2$ | 420 | $\mathrm{J}$ | atkins | e2.1(a)(b) | scibench | [
{
"content": "<|im_start|>system\nA conversation between User and Assistant. The user asks a question, and the assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and<answer> </answer> tags, respectively, i.e., <think> reasoning process here </think><answer> answer here </answer>.\n<|im_end|>\n<|im_start|>user\nAssume all gases are perfect unless stated otherwise. Unless otherwise stated, thermodynamic data are for 298.15 K. Calculate the work needed for a $65 \\mathrm{~kg}$ person to climb through $4.0 \\mathrm{~m}$ on the surface of the Moon $\\left(g=1.60 \\mathrm{~m} \\mathrm{~s}^{-2}\\right)$.\n<|im_end|>\n<|im_start|>assistant\n<think>",
"role": "user"
}
]
| logic | {
"ground_truth": {
"answer_number": "420",
"unit": " $\\mathrm{J}$"
},
"style": "rule"
} | {
"index": 307,
"split": "train"
} |
||
Assume all gases are perfect unless stated otherwise. Unless otherwise stated, thermodynamic data are for 298.15 K. When $120 \mathrm{mg}$ of naphthalene, $\mathrm{C}_{10} \mathrm{H}_8(\mathrm{~s})$, was burned in a bomb calorimeter the temperature rose by $3.05 \mathrm{~K}$. By how much will the temperature rise when $10 \mathrm{mg}$ of phenol, $\mathrm{C}_6 \mathrm{H}_5 \mathrm{OH}(\mathrm{s})$, is burned in the calorimeter under the same conditions? | 205 | 205 | $\text{K}$ | atkins | e2.20(a)(b) | scibench | [
{
"content": "<|im_start|>system\nA conversation between User and Assistant. The user asks a question, and the assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and<answer> </answer> tags, respectively, i.e., <think> reasoning process here </think><answer> answer here </answer>.\n<|im_end|>\n<|im_start|>user\nAssume all gases are perfect unless stated otherwise. Unless otherwise stated, thermodynamic data are for 298.15 K. When $120 \\mathrm{mg}$ of naphthalene, $\\mathrm{C}_{10} \\mathrm{H}_8(\\mathrm{~s})$, was burned in a bomb calorimeter the temperature rose by $3.05 \\mathrm{~K}$. By how much will the temperature rise when $10 \\mathrm{mg}$ of phenol, $\\mathrm{C}_6 \\mathrm{H}_5 \\mathrm{OH}(\\mathrm{s})$, is burned in the calorimeter under the same conditions?\n<|im_end|>\n<|im_start|>assistant\n<think>",
"role": "user"
}
]
| logic | {
"ground_truth": {
"answer_number": "205",
"unit": "$\\text{K}$ "
},
"style": "rule"
} | {
"index": 308,
"split": "train"
} |
||
Assume all gases are perfect unless stated otherwise. Unless otherwise stated, thermodynamic data are for 298.15 K. Calculate the final pressure of a sample of carbon dioxide that expands reversibly and adiabatically from $57.4 \mathrm{kPa}$ and $1.0 \mathrm{dm}^3$ to a final volume of $2.0 \mathrm{dm}^3$. Take $\gamma=1.4$. | 22 | 22 | $\text{kPA}$ | atkins | e2.11(a) | scibench | [
{
"content": "<|im_start|>system\nA conversation between User and Assistant. The user asks a question, and the assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and<answer> </answer> tags, respectively, i.e., <think> reasoning process here </think><answer> answer here </answer>.\n<|im_end|>\n<|im_start|>user\nAssume all gases are perfect unless stated otherwise. Unless otherwise stated, thermodynamic data are for 298.15 K. Calculate the final pressure of a sample of carbon dioxide that expands reversibly and adiabatically from $57.4 \\mathrm{kPa}$ and $1.0 \\mathrm{dm}^3$ to a final volume of $2.0 \\mathrm{dm}^3$. Take $\\gamma=1.4$.\n<|im_end|>\n<|im_start|>assistant\n<think>",
"role": "user"
}
]
| logic | {
"ground_truth": {
"answer_number": "22",
"unit": "$\\text{kPA}$"
},
"style": "rule"
} | {
"index": 309,
"split": "train"
} |
||
Assume all gases are perfect unless stated otherwise. Unless otherwise stated, thermodynamic data are for 298.15 K. When $229 \mathrm{~J}$ of energy is supplied as heat to $3.0 \mathrm{~mol} \mathrm{Ar}(\mathrm{g})$ at constant pressure, the temperature of the sample increases by $2.55 \mathrm{~K}$. Calculate the molar heat capacities at constant volume. | 30 | 30 | $\mathrm{J} \mathrm{K}^{-1} \mathrm{~mol}^{-1}$ | atkins | e2.12(a)(a) | scibench | [
{
"content": "<|im_start|>system\nA conversation between User and Assistant. The user asks a question, and the assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and<answer> </answer> tags, respectively, i.e., <think> reasoning process here </think><answer> answer here </answer>.\n<|im_end|>\n<|im_start|>user\nAssume all gases are perfect unless stated otherwise. Unless otherwise stated, thermodynamic data are for 298.15 K. When $229 \\mathrm{~J}$ of energy is supplied as heat to $3.0 \\mathrm{~mol} \\mathrm{Ar}(\\mathrm{g})$ at constant pressure, the temperature of the sample increases by $2.55 \\mathrm{~K}$. Calculate the molar heat capacities at constant volume.\n<|im_end|>\n<|im_start|>assistant\n<think>",
"role": "user"
}
]
| logic | {
"ground_truth": {
"answer_number": "30",
"unit": "$\\mathrm{J} \\mathrm{K}^{-1} \\mathrm{~mol}^{-1}$"
},
"style": "rule"
} | {
"index": 310,
"split": "train"
} |
||
Assume that all gases are perfect and that data refer to 298.15 K unless otherwise stated. The fugacity coefficient of a certain gas at $200 \mathrm{~K}$ and 50 bar is 0.72. Calculate the difference of its molar Gibbs energy from that of a perfect gas in the same state. | -0.55 | -0.55 | $\mathrm{kJ} \mathrm{mol}^{-1}$ | atkins | e3.20(a) | scibench | [
{
"content": "<|im_start|>system\nA conversation between User and Assistant. The user asks a question, and the assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and<answer> </answer> tags, respectively, i.e., <think> reasoning process here </think><answer> answer here </answer>.\n<|im_end|>\n<|im_start|>user\nAssume that all gases are perfect and that data refer to 298.15 K unless otherwise stated. The fugacity coefficient of a certain gas at $200 \\mathrm{~K}$ and 50 bar is 0.72. Calculate the difference of its molar Gibbs energy from that of a perfect gas in the same state.\n<|im_end|>\n<|im_start|>assistant\n<think>",
"role": "user"
}
]
| logic | {
"ground_truth": {
"answer_number": "-0.55",
"unit": "$\\mathrm{kJ} \\mathrm{mol}^{-1}$"
},
"style": "rule"
} | {
"index": 311,
"split": "train"
} |
||
Assume all gases are perfect unless stated otherwise. Unless otherwise stated, thermodynamic data are for 298.15 K. Given the reactions (1) and (2) below, determine $\Delta_{\mathrm{r}} U^{\ominus}$ for reaction (3).
(1) $\mathrm{H}_2(\mathrm{g})+\mathrm{Cl}_2(\mathrm{g}) \rightarrow 2 \mathrm{HCl}(\mathrm{g})$, $\Delta_{\mathrm{r}} H^{\ominus}=-184.62 \mathrm{~kJ} \mathrm{~mol}^{-1}$
(2) $2 \mathrm{H}_2(\mathrm{g})+\mathrm{O}_2(\mathrm{g}) \rightarrow 2 \mathrm{H}_2 \mathrm{O}(\mathrm{g})$, $\Delta_{\mathrm{r}} H^\ominus=-483.64 \mathrm{~kJ} \mathrm{~mol}^{-1}$
(3) $4 \mathrm{HCl}(\mathrm{g})+\mathrm{O}_2(\mathrm{g}) \rightarrow 2 \mathrm{Cl}_2(\mathrm{g})+2 \mathrm{H}_2 \mathrm{O}(\mathrm{g})$ | -111.92 | -111.92 | $\mathrm{kJ} \mathrm{mol}^{-1}$ | atkins | e2.23(a)(b) | scibench | [
{
"content": "<|im_start|>system\nA conversation between User and Assistant. The user asks a question, and the assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and<answer> </answer> tags, respectively, i.e., <think> reasoning process here </think><answer> answer here </answer>.\n<|im_end|>\n<|im_start|>user\nAssume all gases are perfect unless stated otherwise. Unless otherwise stated, thermodynamic data are for 298.15 K. Given the reactions (1) and (2) below, determine $\\Delta_{\\mathrm{r}} U^{\\ominus}$ for reaction (3).\r\n\r\n(1) $\\mathrm{H}_2(\\mathrm{g})+\\mathrm{Cl}_2(\\mathrm{g}) \\rightarrow 2 \\mathrm{HCl}(\\mathrm{g})$, $\\Delta_{\\mathrm{r}} H^{\\ominus}=-184.62 \\mathrm{~kJ} \\mathrm{~mol}^{-1}$\r\n\r\n(2) $2 \\mathrm{H}_2(\\mathrm{g})+\\mathrm{O}_2(\\mathrm{g}) \\rightarrow 2 \\mathrm{H}_2 \\mathrm{O}(\\mathrm{g})$, $\\Delta_{\\mathrm{r}} H^\\ominus=-483.64 \\mathrm{~kJ} \\mathrm{~mol}^{-1}$\r\n\r\n(3) $4 \\mathrm{HCl}(\\mathrm{g})+\\mathrm{O}_2(\\mathrm{g}) \\rightarrow 2 \\mathrm{Cl}_2(\\mathrm{g})+2 \\mathrm{H}_2 \\mathrm{O}(\\mathrm{g})$\n<|im_end|>\n<|im_start|>assistant\n<think>",
"role": "user"
}
]
| logic | {
"ground_truth": {
"answer_number": "-111.92",
"unit": "$\\mathrm{kJ} \\mathrm{mol}^{-1}$"
},
"style": "rule"
} | {
"index": 312,
"split": "train"
} |
||
A fluid has density $870 \mathrm{~kg} / \mathrm{m}^3$ and flows with velocity $\mathbf{v}=z \mathbf{i}+y^2 \mathbf{j}+x^2 \mathbf{k}$, where $x, y$, and $z$ are measured in meters and the components of $\mathbf{v}$ in meters per second. Find the rate of flow outward through the cylinder $x^2+y^2=4$, $0 \leqslant z \leqslant 1$.
| 0 | 0 | $\mathrm{kg}/\mathrm{s}$ | calculus | 16.7.43 | scibench | [
{
"content": "<|im_start|>system\nA conversation between User and Assistant. The user asks a question, and the assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and<answer> </answer> tags, respectively, i.e., <think> reasoning process here </think><answer> answer here </answer>.\n<|im_end|>\n<|im_start|>user\nA fluid has density $870 \\mathrm{~kg} / \\mathrm{m}^3$ and flows with velocity $\\mathbf{v}=z \\mathbf{i}+y^2 \\mathbf{j}+x^2 \\mathbf{k}$, where $x, y$, and $z$ are measured in meters and the components of $\\mathbf{v}$ in meters per second. Find the rate of flow outward through the cylinder $x^2+y^2=4$, $0 \\leqslant z \\leqslant 1$.\r\n\n<|im_end|>\n<|im_start|>assistant\n<think>",
"role": "user"
}
]
| logic | {
"ground_truth": {
"answer_number": "0",
"unit": " $\\mathrm{kg}/\\mathrm{s}$"
},
"style": "rule"
} | {
"index": 313,
"split": "train"
} |
||
Suppose that $2 \mathrm{~J}$ of work is needed to stretch a spring from its natural length of $30 \mathrm{~cm}$ to a length of $42 \mathrm{~cm}$. How far beyond its natural length will a force of $30 \mathrm{~N}$ keep the spring stretched? | 10.8 | 10.8 | $\mathrm{cm}$ | calculus | 6.4.9(b) | scibench | [
{
"content": "<|im_start|>system\nA conversation between User and Assistant. The user asks a question, and the assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and<answer> </answer> tags, respectively, i.e., <think> reasoning process here </think><answer> answer here </answer>.\n<|im_end|>\n<|im_start|>user\nSuppose that $2 \\mathrm{~J}$ of work is needed to stretch a spring from its natural length of $30 \\mathrm{~cm}$ to a length of $42 \\mathrm{~cm}$. How far beyond its natural length will a force of $30 \\mathrm{~N}$ keep the spring stretched?\n<|im_end|>\n<|im_start|>assistant\n<think>",
"role": "user"
}
]
| logic | {
"ground_truth": {
"answer_number": "10.8",
"unit": " $\\mathrm{cm}$"
},
"style": "rule"
} | {
"index": 314,
"split": "train"
} |
||
Find the work done by a force $\mathbf{F}=8 \mathbf{i}-6 \mathbf{j}+9 \mathbf{k}$ that moves an object from the point $(0,10,8)$ to the point $(6,12,20)$ along a straight line. The distance is measured in meters and the force in newtons. | 144 | 144 | $\mathrm{J}$ | calculus | 12.3.49 | scibench | [
{
"content": "<|im_start|>system\nA conversation between User and Assistant. The user asks a question, and the assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and<answer> </answer> tags, respectively, i.e., <think> reasoning process here </think><answer> answer here </answer>.\n<|im_end|>\n<|im_start|>user\nFind the work done by a force $\\mathbf{F}=8 \\mathbf{i}-6 \\mathbf{j}+9 \\mathbf{k}$ that moves an object from the point $(0,10,8)$ to the point $(6,12,20)$ along a straight line. The distance is measured in meters and the force in newtons.\n<|im_end|>\n<|im_start|>assistant\n<think>",
"role": "user"
}
]
| logic | {
"ground_truth": {
"answer_number": "144",
"unit": " $\\mathrm{J}$"
},
"style": "rule"
} | {
"index": 315,
"split": "train"
} |
||
A ball is thrown at an angle of $45^{\circ}$ to the ground. If the ball lands $90 \mathrm{~m}$ away, what was the initial speed of the ball?
| 30 | 30 | $\mathrm{m}/\mathrm{s}$ | calculus | 13.4.25 | scibench | [
{
"content": "<|im_start|>system\nA conversation between User and Assistant. The user asks a question, and the assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and<answer> </answer> tags, respectively, i.e., <think> reasoning process here </think><answer> answer here </answer>.\n<|im_end|>\n<|im_start|>user\nA ball is thrown at an angle of $45^{\\circ}$ to the ground. If the ball lands $90 \\mathrm{~m}$ away, what was the initial speed of the ball?\r\n\n<|im_end|>\n<|im_start|>assistant\n<think>",
"role": "user"
}
]
| logic | {
"ground_truth": {
"answer_number": "30",
"unit": " $\\mathrm{m}/\\mathrm{s}$"
},
"style": "rule"
} | {
"index": 316,
"split": "train"
} |
||
Show how to approximate the required work by a Riemann sum. Then express the work as an integral and evaluate it. A leaky 10-kg bucket is lifted from the ground to a height of $12 \mathrm{~m}$ at a constant speed with a rope that weighs $0.8 \mathrm{~kg} / \mathrm{m}$. Initially the bucket contains $36 \mathrm{~kg}$ of water, but the water leaks at a constant rate and finishes draining just as the bucket reaches the 12-m level. How much work is done?
| 3857 | 3857 | $\mathrm{J}$ | calculus | 6.4.17 | scibench | [
{
"content": "<|im_start|>system\nA conversation between User and Assistant. The user asks a question, and the assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and<answer> </answer> tags, respectively, i.e., <think> reasoning process here </think><answer> answer here </answer>.\n<|im_end|>\n<|im_start|>user\nShow how to approximate the required work by a Riemann sum. Then express the work as an integral and evaluate it. A leaky 10-kg bucket is lifted from the ground to a height of $12 \\mathrm{~m}$ at a constant speed with a rope that weighs $0.8 \\mathrm{~kg} / \\mathrm{m}$. Initially the bucket contains $36 \\mathrm{~kg}$ of water, but the water leaks at a constant rate and finishes draining just as the bucket reaches the 12-m level. How much work is done?\r\n\n<|im_end|>\n<|im_start|>assistant\n<think>",
"role": "user"
}
]
| logic | {
"ground_truth": {
"answer_number": "3857",
"unit": " $\\mathrm{J}$"
},
"style": "rule"
} | {
"index": 317,
"split": "train"
} |
||
Find the volume of the described solid $S$. The base of $S$ is an elliptical region with boundary curve $9 x^2+4 y^2=36$. Cross-sections perpendicular to the $x$-axis are isosceles right triangles with hypotenuse in the base. | 24 | 24 | calculus | 6.2.55 | scibench | [
{
"content": "<|im_start|>system\nA conversation between User and Assistant. The user asks a question, and the assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and<answer> </answer> tags, respectively, i.e., <think> reasoning process here </think><answer> answer here </answer>.\n<|im_end|>\n<|im_start|>user\nFind the volume of the described solid $S$. The base of $S$ is an elliptical region with boundary curve $9 x^2+4 y^2=36$. Cross-sections perpendicular to the $x$-axis are isosceles right triangles with hypotenuse in the base.\n<|im_end|>\n<|im_start|>assistant\n<think>",
"role": "user"
}
]
| logic | {
"ground_truth": {
"answer_number": "24",
"unit": " "
},
"style": "rule"
} | {
"index": 318,
"split": "train"
} |
|||
A swimming pool is circular with a $40-\mathrm{ft}$ diameter. The depth is constant along east-west lines and increases linearly from $2 \mathrm{ft}$ at the south end to $7 \mathrm{ft}$ at the north end. Find the volume of water in the pool. | $1800\pi$ | 5654.86677646 | $\mathrm{ft}^3$ | calculus | 15.4.35 | scibench | [
{
"content": "<|im_start|>system\nA conversation between User and Assistant. The user asks a question, and the assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and<answer> </answer> tags, respectively, i.e., <think> reasoning process here </think><answer> answer here </answer>.\n<|im_end|>\n<|im_start|>user\nA swimming pool is circular with a $40-\\mathrm{ft}$ diameter. The depth is constant along east-west lines and increases linearly from $2 \\mathrm{ft}$ at the south end to $7 \\mathrm{ft}$ at the north end. Find the volume of water in the pool.\n<|im_end|>\n<|im_start|>assistant\n<think>",
"role": "user"
}
]
| logic | {
"ground_truth": {
"answer_number": "5654.86677646",
"unit": " $\\mathrm{ft}^3$"
},
"style": "rule"
} | {
"index": 319,
"split": "train"
} |
||
The orbit of Halley's comet, last seen in 1986 and due to return in 2062, is an ellipse with eccentricity 0.97 and one focus at the sun. The length of its major axis is $36.18 \mathrm{AU}$. [An astronomical unit (AU) is the mean distance between the earth and the sun, about 93 million miles.] By finding a polar equation for the orbit of Halley's comet, what is the maximum distance from the comet to the sun? | 35.64 | 35.64 | $\mathrm{AU}$ | calculus | 10.6.27 | scibench | [
{
"content": "<|im_start|>system\nA conversation between User and Assistant. The user asks a question, and the assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and<answer> </answer> tags, respectively, i.e., <think> reasoning process here </think><answer> answer here </answer>.\n<|im_end|>\n<|im_start|>user\nThe orbit of Halley's comet, last seen in 1986 and due to return in 2062, is an ellipse with eccentricity 0.97 and one focus at the sun. The length of its major axis is $36.18 \\mathrm{AU}$. [An astronomical unit (AU) is the mean distance between the earth and the sun, about 93 million miles.] By finding a polar equation for the orbit of Halley's comet, what is the maximum distance from the comet to the sun?\n<|im_end|>\n<|im_start|>assistant\n<think>",
"role": "user"
}
]
| logic | {
"ground_truth": {
"answer_number": "35.64",
"unit": " $\\mathrm{AU}$"
},
"style": "rule"
} | {
"index": 320,
"split": "train"
} |
||
If a ball is thrown into the air with a velocity of $40 \mathrm{ft} / \mathrm{s}$, its height in feet $t$ seconds later is given by $y=40 t-16 t^2$. Find the average velocity for the time period beginning when $t=2$ and lasting 0.5 second. | -32 | -32 | $\mathrm{ft} / \mathrm{s}$ | calculus | 2.1.5(a) | scibench | [
{
"content": "<|im_start|>system\nA conversation between User and Assistant. The user asks a question, and the assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and<answer> </answer> tags, respectively, i.e., <think> reasoning process here </think><answer> answer here </answer>.\n<|im_end|>\n<|im_start|>user\n If a ball is thrown into the air with a velocity of $40 \\mathrm{ft} / \\mathrm{s}$, its height in feet $t$ seconds later is given by $y=40 t-16 t^2$. Find the average velocity for the time period beginning when $t=2$ and lasting 0.5 second.\n<|im_end|>\n<|im_start|>assistant\n<think>",
"role": "user"
}
]
| logic | {
"ground_truth": {
"answer_number": "-32",
"unit": "$\\mathrm{ft} / \\mathrm{s}$"
},
"style": "rule"
} | {
"index": 321,
"split": "train"
} |
||
A CAT scan produces equally spaced cross-sectional views of a human organ that provide information about the organ otherwise obtained only by surgery. Suppose that a CAT scan of a human liver shows cross-sections spaced $1.5 \mathrm{~cm}$ apart. The liver is $15 \mathrm{~cm}$ long and the cross-sectional areas, in square centimeters, are $0,18,58,79,94,106,117,128,63, 39, 0$. Use the Midpoint Rule to estimate the volume of the liver.
| 1110 | 1110 | $\mathrm{cm}^3$ | calculus | 6.2.43 | scibench | [
{
"content": "<|im_start|>system\nA conversation between User and Assistant. The user asks a question, and the assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and<answer> </answer> tags, respectively, i.e., <think> reasoning process here </think><answer> answer here </answer>.\n<|im_end|>\n<|im_start|>user\nA CAT scan produces equally spaced cross-sectional views of a human organ that provide information about the organ otherwise obtained only by surgery. Suppose that a CAT scan of a human liver shows cross-sections spaced $1.5 \\mathrm{~cm}$ apart. The liver is $15 \\mathrm{~cm}$ long and the cross-sectional areas, in square centimeters, are $0,18,58,79,94,106,117,128,63, 39, 0$. Use the Midpoint Rule to estimate the volume of the liver.\r\n\n<|im_end|>\n<|im_start|>assistant\n<think>",
"role": "user"
}
]
| logic | {
"ground_truth": {
"answer_number": "1110",
"unit": " $\\mathrm{cm}^3$"
},
"style": "rule"
} | {
"index": 322,
"split": "train"
} |
||
A manufacturer of corrugated metal roofing wants to produce panels that are $28 \mathrm{in}$. wide and $2 \mathrm{in}$. thick by processing flat sheets of metal as shown in the figure. The profile of the roofing takes the shape of a sine wave. Verify that the sine curve has equation $y=\sin (\pi x / 7)$ and find the width $w$ of a flat metal sheet that is needed to make a 28-inch panel. (Use your calculator to evaluate the integral correct to four significant digits.)
| 29.36 | 29.36 | $\mathrm{in}$ | calculus | 8.1.39 | scibench | [
{
"content": "<|im_start|>system\nA conversation between User and Assistant. The user asks a question, and the assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and<answer> </answer> tags, respectively, i.e., <think> reasoning process here </think><answer> answer here </answer>.\n<|im_end|>\n<|im_start|>user\nA manufacturer of corrugated metal roofing wants to produce panels that are $28 \\mathrm{in}$. wide and $2 \\mathrm{in}$. thick by processing flat sheets of metal as shown in the figure. The profile of the roofing takes the shape of a sine wave. Verify that the sine curve has equation $y=\\sin (\\pi x / 7)$ and find the width $w$ of a flat metal sheet that is needed to make a 28-inch panel. (Use your calculator to evaluate the integral correct to four significant digits.)\r\n\n<|im_end|>\n<|im_start|>assistant\n<think>",
"role": "user"
}
]
| logic | {
"ground_truth": {
"answer_number": "29.36",
"unit": " $\\mathrm{in}$"
},
"style": "rule"
} | {
"index": 323,
"split": "train"
} |
||
The dye dilution method is used to measure cardiac output with $6 \mathrm{mg}$ of dye. The dye concentrations, in $\mathrm{mg} / \mathrm{L}$, are modeled by $c(t)=20 t e^{-0.6 t}, 0 \leqslant t \leqslant 10$, where $t$ is measured in seconds. Find the cardiac output. | 6.6 | 6.6 | $\mathrm{L}/\mathrm{min}$ | calculus | 8.4.17 | scibench | [
{
"content": "<|im_start|>system\nA conversation between User and Assistant. The user asks a question, and the assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and<answer> </answer> tags, respectively, i.e., <think> reasoning process here </think><answer> answer here </answer>.\n<|im_end|>\n<|im_start|>user\nThe dye dilution method is used to measure cardiac output with $6 \\mathrm{mg}$ of dye. The dye concentrations, in $\\mathrm{mg} / \\mathrm{L}$, are modeled by $c(t)=20 t e^{-0.6 t}, 0 \\leqslant t \\leqslant 10$, where $t$ is measured in seconds. Find the cardiac output.\n<|im_end|>\n<|im_start|>assistant\n<think>",
"role": "user"
}
]
| logic | {
"ground_truth": {
"answer_number": "6.6",
"unit": " $\\mathrm{L}/\\mathrm{min}$"
},
"style": "rule"
} | {
"index": 324,
"split": "train"
} |
||
A planning engineer for a new alum plant must present some estimates to his company regarding the capacity of a silo designed to contain bauxite ore until it is processed into alum. The ore resembles pink talcum powder and is poured from a conveyor at the top of the silo. The silo is a cylinder $100 \mathrm{ft}$ high with a radius of $200 \mathrm{ft}$. The conveyor carries ore at a rate of $60,000 \pi \mathrm{~ft}^3 / \mathrm{h}$ and the ore maintains a conical shape whose radius is 1.5 times its height. If, at a certain time $t$, the pile is $60 \mathrm{ft}$ high, how long will it take for the pile to reach the top of the silo? | 9.8 | 9.8 | $\mathrm{h}$ | calculus | 9.RP.11(a) | Review Plus Problem | scibench | [
{
"content": "<|im_start|>system\nA conversation between User and Assistant. The user asks a question, and the assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and<answer> </answer> tags, respectively, i.e., <think> reasoning process here </think><answer> answer here </answer>.\n<|im_end|>\n<|im_start|>user\nA planning engineer for a new alum plant must present some estimates to his company regarding the capacity of a silo designed to contain bauxite ore until it is processed into alum. The ore resembles pink talcum powder and is poured from a conveyor at the top of the silo. The silo is a cylinder $100 \\mathrm{ft}$ high with a radius of $200 \\mathrm{ft}$. The conveyor carries ore at a rate of $60,000 \\pi \\mathrm{~ft}^3 / \\mathrm{h}$ and the ore maintains a conical shape whose radius is 1.5 times its height. If, at a certain time $t$, the pile is $60 \\mathrm{ft}$ high, how long will it take for the pile to reach the top of the silo?\n<|im_end|>\n<|im_start|>assistant\n<think>",
"role": "user"
}
]
| logic | {
"ground_truth": {
"answer_number": "9.8",
"unit": " $\\mathrm{h}$"
},
"style": "rule"
} | {
"index": 325,
"split": "train"
} |
|
A boatman wants to cross a canal that is $3 \mathrm{~km}$ wide and wants to land at a point $2 \mathrm{~km}$ upstream from his starting point. The current in the canal flows at $3.5 \mathrm{~km} / \mathrm{h}$ and the speed of his boat is $13 \mathrm{~km} / \mathrm{h}$. How long will the trip take?
| 20.2 | 20.2 | $\mathrm{min}$ | calculus | 12.2.39 | scibench | [
{
"content": "<|im_start|>system\nA conversation between User and Assistant. The user asks a question, and the assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and<answer> </answer> tags, respectively, i.e., <think> reasoning process here </think><answer> answer here </answer>.\n<|im_end|>\n<|im_start|>user\nA boatman wants to cross a canal that is $3 \\mathrm{~km}$ wide and wants to land at a point $2 \\mathrm{~km}$ upstream from his starting point. The current in the canal flows at $3.5 \\mathrm{~km} / \\mathrm{h}$ and the speed of his boat is $13 \\mathrm{~km} / \\mathrm{h}$. How long will the trip take?\r\n\n<|im_end|>\n<|im_start|>assistant\n<think>",
"role": "user"
}
]
| logic | {
"ground_truth": {
"answer_number": "20.2",
"unit": " $\\mathrm{min}$"
},
"style": "rule"
} | {
"index": 326,
"split": "train"
} |
||
Find the area bounded by the curves $y=\cos x$ and $y=\cos ^2 x$ between $x=0$ and $x=\pi$. | 2 | 2 | calculus | 7.R.73 | scibench | [
{
"content": "<|im_start|>system\nA conversation between User and Assistant. The user asks a question, and the assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and<answer> </answer> tags, respectively, i.e., <think> reasoning process here </think><answer> answer here </answer>.\n<|im_end|>\n<|im_start|>user\nFind the area bounded by the curves $y=\\cos x$ and $y=\\cos ^2 x$ between $x=0$ and $x=\\pi$.\n<|im_end|>\n<|im_start|>assistant\n<think>",
"role": "user"
}
]
| logic | {
"ground_truth": {
"answer_number": "2",
"unit": " "
},
"style": "rule"
} | {
"index": 327,
"split": "train"
} |
|||
A sled is pulled along a level path through snow by a rope. A 30-lb force acting at an angle of $40^{\circ}$ above the horizontal moves the sled $80 \mathrm{ft}$. Find the work done by the force. | $2400\cos({40}^{\circ})$ | 1838.50666349 | $\mathrm{ft-lb}$ | calculus | 12.3.51 | scibench | [
{
"content": "<|im_start|>system\nA conversation between User and Assistant. The user asks a question, and the assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and<answer> </answer> tags, respectively, i.e., <think> reasoning process here </think><answer> answer here </answer>.\n<|im_end|>\n<|im_start|>user\nA sled is pulled along a level path through snow by a rope. A 30-lb force acting at an angle of $40^{\\circ}$ above the horizontal moves the sled $80 \\mathrm{ft}$. Find the work done by the force.\n<|im_end|>\n<|im_start|>assistant\n<think>",
"role": "user"
}
]
| logic | {
"ground_truth": {
"answer_number": "1838.50666349",
"unit": " $\\mathrm{ft-lb}$"
},
"style": "rule"
} | {
"index": 328,
"split": "train"
} |
||
If $R$ is the total resistance of three resistors, connected in parallel, with resistances $R_1, R_2, R_3$, then
$$
\frac{1}{R}=\frac{1}{R_1}+\frac{1}{R_2}+\frac{1}{R_3}
$$
If the resistances are measured in ohms as $R_1=25 \Omega$, $R_2=40 \Omega$, and $R_3=50 \Omega$, with a possible error of $0.5 \%$ in each case, estimate the maximum error in the calculated value of $R$. | $\frac{1}{17}$ | 0.05882352941 | $\Omega$ | calculus | 14.4.39 | scibench | [
{
"content": "<|im_start|>system\nA conversation between User and Assistant. The user asks a question, and the assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and<answer> </answer> tags, respectively, i.e., <think> reasoning process here </think><answer> answer here </answer>.\n<|im_end|>\n<|im_start|>user\n If $R$ is the total resistance of three resistors, connected in parallel, with resistances $R_1, R_2, R_3$, then\r\n$$\r\n\\frac{1}{R}=\\frac{1}{R_1}+\\frac{1}{R_2}+\\frac{1}{R_3}\r\n$$\r\nIf the resistances are measured in ohms as $R_1=25 \\Omega$, $R_2=40 \\Omega$, and $R_3=50 \\Omega$, with a possible error of $0.5 \\%$ in each case, estimate the maximum error in the calculated value of $R$.\n<|im_end|>\n<|im_start|>assistant\n<think>",
"role": "user"
}
]
| logic | {
"ground_truth": {
"answer_number": "0.05882352941",
"unit": " $\\Omega$"
},
"style": "rule"
} | {
"index": 329,
"split": "train"
} |
||
The length and width of a rectangle are measured as $30 \mathrm{~cm}$ and $24 \mathrm{~cm}$, respectively, with an error in measurement of at most $0.1 \mathrm{~cm}$ in each. Use differentials to estimate the maximum error in the calculated area of the rectangle.
| 5.4 | 5.4 | $\mathrm{cm^2}$ | calculus | 14.4.33 | scibench | [
{
"content": "<|im_start|>system\nA conversation between User and Assistant. The user asks a question, and the assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and<answer> </answer> tags, respectively, i.e., <think> reasoning process here </think><answer> answer here </answer>.\n<|im_end|>\n<|im_start|>user\nThe length and width of a rectangle are measured as $30 \\mathrm{~cm}$ and $24 \\mathrm{~cm}$, respectively, with an error in measurement of at most $0.1 \\mathrm{~cm}$ in each. Use differentials to estimate the maximum error in the calculated area of the rectangle.\r\n\n<|im_end|>\n<|im_start|>assistant\n<think>",
"role": "user"
}
]
| logic | {
"ground_truth": {
"answer_number": "5.4",
"unit": " $\\mathrm{cm^2}$"
},
"style": "rule"
} | {
"index": 330,
"split": "train"
} |
||
The planet Mercury travels in an elliptical orbit with eccentricity 0.206 . Its minimum distance from the sun is $4.6 \times 10^7 \mathrm{~km}$. Find its maximum distance from the sun. | 7 | 7 | $\mathrm{10^7} \mathrm{~km}$ | calculus | 10.6.29 | scibench | [
{
"content": "<|im_start|>system\nA conversation between User and Assistant. The user asks a question, and the assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and<answer> </answer> tags, respectively, i.e., <think> reasoning process here </think><answer> answer here </answer>.\n<|im_end|>\n<|im_start|>user\nThe planet Mercury travels in an elliptical orbit with eccentricity 0.206 . Its minimum distance from the sun is $4.6 \\times 10^7 \\mathrm{~km}$. Find its maximum distance from the sun.\n<|im_end|>\n<|im_start|>assistant\n<think>",
"role": "user"
}
]
| logic | {
"ground_truth": {
"answer_number": "7",
"unit": " $\\mathrm{10^7} \\mathrm{~km}$"
},
"style": "rule"
} | {
"index": 331,
"split": "train"
} |
||
Use differentials to estimate the amount of tin in a closed tin can with diameter $8 \mathrm{~cm}$ and height $12 \mathrm{~cm}$ if the tin is $0.04 \mathrm{~cm}$ thick. | 16 | 16 | $\mathrm{cm^3}$ | calculus | 14.4.35 | scibench | [
{
"content": "<|im_start|>system\nA conversation between User and Assistant. The user asks a question, and the assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and<answer> </answer> tags, respectively, i.e., <think> reasoning process here </think><answer> answer here </answer>.\n<|im_end|>\n<|im_start|>user\nUse differentials to estimate the amount of tin in a closed tin can with diameter $8 \\mathrm{~cm}$ and height $12 \\mathrm{~cm}$ if the tin is $0.04 \\mathrm{~cm}$ thick.\n<|im_end|>\n<|im_start|>assistant\n<think>",
"role": "user"
}
]
| logic | {
"ground_truth": {
"answer_number": "16",
"unit": " $\\mathrm{cm^3}$"
},
"style": "rule"
} | {
"index": 332,
"split": "train"
} |
||
Show how to approximate the required work by a Riemann sum. Then express the work as an integral and evaluate it. A cable that weighs $2 \mathrm{~lb} / \mathrm{ft}$ is used to lift $800 \mathrm{~lb}$ of coal up a mine shaft $500 \mathrm{~ft}$ deep. Find the work done.
| 650000 | 650000 | $\mathrm{ft-lb}$ | calculus | 6.4.15 | scibench | [
{
"content": "<|im_start|>system\nA conversation between User and Assistant. The user asks a question, and the assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and<answer> </answer> tags, respectively, i.e., <think> reasoning process here </think><answer> answer here </answer>.\n<|im_end|>\n<|im_start|>user\nShow how to approximate the required work by a Riemann sum. Then express the work as an integral and evaluate it. A cable that weighs $2 \\mathrm{~lb} / \\mathrm{ft}$ is used to lift $800 \\mathrm{~lb}$ of coal up a mine shaft $500 \\mathrm{~ft}$ deep. Find the work done.\r\n\n<|im_end|>\n<|im_start|>assistant\n<think>",
"role": "user"
}
]
| logic | {
"ground_truth": {
"answer_number": "650000",
"unit": " $\\mathrm{ft-lb}$"
},
"style": "rule"
} | {
"index": 333,
"split": "train"
} |
||
A patient takes $150 \mathrm{mg}$ of a drug at the same time every day. Just before each tablet is taken, 5$\%$ of the drug remains in the body. What quantity of the drug is in the body after the third tablet? | 157.875 | 157.875 | $\mathrm{mg}$ | calculus | 11.2.69(a) | scibench | [
{
"content": "<|im_start|>system\nA conversation between User and Assistant. The user asks a question, and the assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and<answer> </answer> tags, respectively, i.e., <think> reasoning process here </think><answer> answer here </answer>.\n<|im_end|>\n<|im_start|>user\n A patient takes $150 \\mathrm{mg}$ of a drug at the same time every day. Just before each tablet is taken, 5$\\%$ of the drug remains in the body. What quantity of the drug is in the body after the third tablet? \n<|im_end|>\n<|im_start|>assistant\n<think>",
"role": "user"
}
]
| logic | {
"ground_truth": {
"answer_number": "157.875",
"unit": " $\\mathrm{mg}$"
},
"style": "rule"
} | {
"index": 334,
"split": "train"
} |
||
A $360-\mathrm{lb}$ gorilla climbs a tree to a height of $20 \mathrm{~ft}$. Find the work done if the gorilla reaches that height in 5 seconds. | 7200 | 7200 | $\mathrm{ft-lb}$ | calculus | 6.4.1(b) | scibench | [
{
"content": "<|im_start|>system\nA conversation between User and Assistant. The user asks a question, and the assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and<answer> </answer> tags, respectively, i.e., <think> reasoning process here </think><answer> answer here </answer>.\n<|im_end|>\n<|im_start|>user\nA $360-\\mathrm{lb}$ gorilla climbs a tree to a height of $20 \\mathrm{~ft}$. Find the work done if the gorilla reaches that height in 5 seconds.\n<|im_end|>\n<|im_start|>assistant\n<think>",
"role": "user"
}
]
| logic | {
"ground_truth": {
"answer_number": "7200",
"unit": " $\\mathrm{ft-lb}$"
},
"style": "rule"
} | {
"index": 335,
"split": "train"
} |
||
Find the area of triangle $A B C$, correct to five decimal places, if
$$
|A B|=10 \mathrm{~cm} \quad|B C|=3 \mathrm{~cm} \quad \angle A B C=107^{\circ}
$$ | 14.34457 | 14.34457 | $\mathrm{cm^2}$ | calculus | D.89 | scibench | [
{
"content": "<|im_start|>system\nA conversation between User and Assistant. The user asks a question, and the assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and<answer> </answer> tags, respectively, i.e., <think> reasoning process here </think><answer> answer here </answer>.\n<|im_end|>\n<|im_start|>user\nFind the area of triangle $A B C$, correct to five decimal places, if\r\n$$\r\n|A B|=10 \\mathrm{~cm} \\quad|B C|=3 \\mathrm{~cm} \\quad \\angle A B C=107^{\\circ}\r\n$$\n<|im_end|>\n<|im_start|>assistant\n<think>",
"role": "user"
}
]
| logic | {
"ground_truth": {
"answer_number": "14.34457",
"unit": " $\\mathrm{cm^2}$"
},
"style": "rule"
} | {
"index": 336,
"split": "train"
} |
||
Use Stokes' Theorem to evaluate $\int_C \mathbf{F} \cdot d \mathbf{r}$, where $\mathbf{F}(x, y, z)=x y \mathbf{i}+y z \mathbf{j}+z x \mathbf{k}$, and $C$ is the triangle with vertices $(1,0,0),(0,1,0)$, and $(0,0,1)$, oriented counterclockwise as viewed from above.
| $-\frac{1}{2}$ | -0.5 | calculus | 16.R.33 | scibench | [
{
"content": "<|im_start|>system\nA conversation between User and Assistant. The user asks a question, and the assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and<answer> </answer> tags, respectively, i.e., <think> reasoning process here </think><answer> answer here </answer>.\n<|im_end|>\n<|im_start|>user\nUse Stokes' Theorem to evaluate $\\int_C \\mathbf{F} \\cdot d \\mathbf{r}$, where $\\mathbf{F}(x, y, z)=x y \\mathbf{i}+y z \\mathbf{j}+z x \\mathbf{k}$, and $C$ is the triangle with vertices $(1,0,0),(0,1,0)$, and $(0,0,1)$, oriented counterclockwise as viewed from above.\r\n\n<|im_end|>\n<|im_start|>assistant\n<think>",
"role": "user"
}
]
| logic | {
"ground_truth": {
"answer_number": "-0.5",
"unit": " "
},
"style": "rule"
} | {
"index": 337,
"split": "train"
} |
|||
A hawk flying at $15 \mathrm{~m} / \mathrm{s}$ at an altitude of $180 \mathrm{~m}$ accidentally drops its prey. The parabolic trajectory of the falling prey is described by the equation
$$
y=180-\frac{x^2}{45}
$$
until it hits the ground, where $y$ is its height above the ground and $x$ is the horizontal distance traveled in meters. Calculate the distance traveled by the prey from the time it is dropped until the time it hits the ground. Express your answer correct to the nearest tenth of a meter. | 209.1 | 209.1 | $\mathrm{m}$ | calculus | 8.1.37 | scibench | [
{
"content": "<|im_start|>system\nA conversation between User and Assistant. The user asks a question, and the assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and<answer> </answer> tags, respectively, i.e., <think> reasoning process here </think><answer> answer here </answer>.\n<|im_end|>\n<|im_start|>user\nA hawk flying at $15 \\mathrm{~m} / \\mathrm{s}$ at an altitude of $180 \\mathrm{~m}$ accidentally drops its prey. The parabolic trajectory of the falling prey is described by the equation\r\n$$\r\ny=180-\\frac{x^2}{45}\r\n$$\r\nuntil it hits the ground, where $y$ is its height above the ground and $x$ is the horizontal distance traveled in meters. Calculate the distance traveled by the prey from the time it is dropped until the time it hits the ground. Express your answer correct to the nearest tenth of a meter.\n<|im_end|>\n<|im_start|>assistant\n<think>",
"role": "user"
}
]
| logic | {
"ground_truth": {
"answer_number": "209.1",
"unit": " $\\mathrm{m}$"
},
"style": "rule"
} | {
"index": 338,
"split": "train"
} |
||
The intensity of light with wavelength $\lambda$ traveling through a diffraction grating with $N$ slits at an angle $\theta$ is given by $I(\theta)=N^2 \sin ^2 k / k^2$, where $k=(\pi N d \sin \theta) / \lambda$ and $d$ is the distance between adjacent slits. A helium-neon laser with wavelength $\lambda=632.8 \times 10^{-9} \mathrm{~m}$ is emitting a narrow band of light, given by $-10^{-6}<\theta<10^{-6}$, through a grating with 10,000 slits spaced $10^{-4} \mathrm{~m}$ apart. Use the Midpoint Rule with $n=10$ to estimate the total light intensity $\int_{-10^{-6}}^{10^{-6}} I(\theta) d \theta$ emerging from the grating. | 59.4 | 59.4 | calculus | 7.7.43 | scibench | [
{
"content": "<|im_start|>system\nA conversation between User and Assistant. The user asks a question, and the assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and<answer> </answer> tags, respectively, i.e., <think> reasoning process here </think><answer> answer here </answer>.\n<|im_end|>\n<|im_start|>user\nThe intensity of light with wavelength $\\lambda$ traveling through a diffraction grating with $N$ slits at an angle $\\theta$ is given by $I(\\theta)=N^2 \\sin ^2 k / k^2$, where $k=(\\pi N d \\sin \\theta) / \\lambda$ and $d$ is the distance between adjacent slits. A helium-neon laser with wavelength $\\lambda=632.8 \\times 10^{-9} \\mathrm{~m}$ is emitting a narrow band of light, given by $-10^{-6}<\\theta<10^{-6}$, through a grating with 10,000 slits spaced $10^{-4} \\mathrm{~m}$ apart. Use the Midpoint Rule with $n=10$ to estimate the total light intensity $\\int_{-10^{-6}}^{10^{-6}} I(\\theta) d \\theta$ emerging from the grating.\n<|im_end|>\n<|im_start|>assistant\n<think>",
"role": "user"
}
]
| logic | {
"ground_truth": {
"answer_number": "59.4",
"unit": " "
},
"style": "rule"
} | {
"index": 339,
"split": "train"
} |
|||
A model for the surface area of a human body is given by $S=0.1091 w^{0.425} h^{0.725}$, where $w$ is the weight (in pounds), $h$ is the height (in inches), and $S$ is measured in square feet. If the errors in measurement of $w$ and $h$ are at most $2 \%$, use differentials to estimate the maximum percentage error in the calculated surface area. | 2.3 | 2.3 | $\%$ | calculus | 14.4.41 | scibench | [
{
"content": "<|im_start|>system\nA conversation between User and Assistant. The user asks a question, and the assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and<answer> </answer> tags, respectively, i.e., <think> reasoning process here </think><answer> answer here </answer>.\n<|im_end|>\n<|im_start|>user\nA model for the surface area of a human body is given by $S=0.1091 w^{0.425} h^{0.725}$, where $w$ is the weight (in pounds), $h$ is the height (in inches), and $S$ is measured in square feet. If the errors in measurement of $w$ and $h$ are at most $2 \\%$, use differentials to estimate the maximum percentage error in the calculated surface area.\n<|im_end|>\n<|im_start|>assistant\n<think>",
"role": "user"
}
]
| logic | {
"ground_truth": {
"answer_number": "2.3",
"unit": " $\\%$"
},
"style": "rule"
} | {
"index": 340,
"split": "train"
} |
||
The temperature at the point $(x, y, z)$ in a substance with conductivity $K=6.5$ is $u(x, y, z)=2 y^2+2 z^2$. Find the rate of heat flow inward across the cylindrical surface $y^2+z^2=6$, $0 \leqslant x \leqslant 4$ | $1248\pi$ | 3920.70763168 | calculus | 16.7.47 | scibench | [
{
"content": "<|im_start|>system\nA conversation between User and Assistant. The user asks a question, and the assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and<answer> </answer> tags, respectively, i.e., <think> reasoning process here </think><answer> answer here </answer>.\n<|im_end|>\n<|im_start|>user\nThe temperature at the point $(x, y, z)$ in a substance with conductivity $K=6.5$ is $u(x, y, z)=2 y^2+2 z^2$. Find the rate of heat flow inward across the cylindrical surface $y^2+z^2=6$, $0 \\leqslant x \\leqslant 4$\n<|im_end|>\n<|im_start|>assistant\n<think>",
"role": "user"
}
]
| logic | {
"ground_truth": {
"answer_number": "3920.70763168",
"unit": " "
},
"style": "rule"
} | {
"index": 341,
"split": "train"
} |
|||
If a ball is thrown into the air with a velocity of $40 \mathrm{ft} / \mathrm{s}$, its height (in feet) after $t$ seconds is given by $y=40 t-16 t^2$. Find the velocity when $t=2$. | -24 | -24 | $\mathrm{ft} / \mathrm{s}$ | calculus | 2.7.13 | scibench | [
{
"content": "<|im_start|>system\nA conversation between User and Assistant. The user asks a question, and the assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and<answer> </answer> tags, respectively, i.e., <think> reasoning process here </think><answer> answer here </answer>.\n<|im_end|>\n<|im_start|>user\nIf a ball is thrown into the air with a velocity of $40 \\mathrm{ft} / \\mathrm{s}$, its height (in feet) after $t$ seconds is given by $y=40 t-16 t^2$. Find the velocity when $t=2$.\n<|im_end|>\n<|im_start|>assistant\n<think>",
"role": "user"
}
]
| logic | {
"ground_truth": {
"answer_number": "-24",
"unit": " $\\mathrm{ft} / \\mathrm{s}$"
},
"style": "rule"
} | {
"index": 342,
"split": "train"
} |
||
A woman walks due west on the deck of a ship at $3 \mathrm{mi} / \mathrm{h}$. The ship is moving north at a speed of $22 \mathrm{mi} / \mathrm{h}$. Find the speed of the woman relative to the surface of the water. | $\sqrt{493}$ | 22.2036033112 | $\mathrm{mi}/\mathrm{h}$ | calculus | 12.2.35 | scibench | [
{
"content": "<|im_start|>system\nA conversation between User and Assistant. The user asks a question, and the assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and<answer> </answer> tags, respectively, i.e., <think> reasoning process here </think><answer> answer here </answer>.\n<|im_end|>\n<|im_start|>user\nA woman walks due west on the deck of a ship at $3 \\mathrm{mi} / \\mathrm{h}$. The ship is moving north at a speed of $22 \\mathrm{mi} / \\mathrm{h}$. Find the speed of the woman relative to the surface of the water.\n<|im_end|>\n<|im_start|>assistant\n<think>",
"role": "user"
}
]
| logic | {
"ground_truth": {
"answer_number": "22.2036033112",
"unit": " $\\mathrm{mi}/\\mathrm{h}$"
},
"style": "rule"
} | {
"index": 343,
"split": "train"
} |
||
A $360-\mathrm{lb}$ gorilla climbs a tree to a height of $20 \mathrm{~ft}$. Find the work done if the gorilla reaches that height in 10 seconds. | 7200 | 7200 | $\mathrm{ft-lb}$ | calculus | 6.4.1(a) | scibench | [
{
"content": "<|im_start|>system\nA conversation between User and Assistant. The user asks a question, and the assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and<answer> </answer> tags, respectively, i.e., <think> reasoning process here </think><answer> answer here </answer>.\n<|im_end|>\n<|im_start|>user\nA $360-\\mathrm{lb}$ gorilla climbs a tree to a height of $20 \\mathrm{~ft}$. Find the work done if the gorilla reaches that height in 10 seconds. \n<|im_end|>\n<|im_start|>assistant\n<think>",
"role": "user"
}
]
| logic | {
"ground_truth": {
"answer_number": "7200",
"unit": "$\\mathrm{ft-lb}$"
},
"style": "rule"
} | {
"index": 344,
"split": "train"
} |
||
A ball is thrown eastward into the air from the origin (in the direction of the positive $x$-axis). The initial velocity is $50 \mathrm{i}+80 \mathrm{k}$, with speed measured in feet per second. The spin of the ball results in a southward acceleration of $4 \mathrm{ft} / \mathrm{s}^2$, so the acceleration vector is $\mathbf{a}=-4 \mathbf{j}-32 \mathbf{k}$. What speed does the ball land?
| $10\sqrt{93}$ | 96.4365076099 | $\mathrm{ft}/\mathrm{s}$ | calculus | 13.4.31 | scibench | [
{
"content": "<|im_start|>system\nA conversation between User and Assistant. The user asks a question, and the assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and<answer> </answer> tags, respectively, i.e., <think> reasoning process here </think><answer> answer here </answer>.\n<|im_end|>\n<|im_start|>user\n A ball is thrown eastward into the air from the origin (in the direction of the positive $x$-axis). The initial velocity is $50 \\mathrm{i}+80 \\mathrm{k}$, with speed measured in feet per second. The spin of the ball results in a southward acceleration of $4 \\mathrm{ft} / \\mathrm{s}^2$, so the acceleration vector is $\\mathbf{a}=-4 \\mathbf{j}-32 \\mathbf{k}$. What speed does the ball land?\r\n\n<|im_end|>\n<|im_start|>assistant\n<think>",
"role": "user"
}
]
| logic | {
"ground_truth": {
"answer_number": "96.4365076099",
"unit": " $\\mathrm{ft}/\\mathrm{s}$"
},
"style": "rule"
} | {
"index": 345,
"split": "train"
} |
||
The demand function for a commodity is given by
$$
p=2000-0.1 x-0.01 x^2
$$
Find the consumer surplus when the sales level is 100 . | 7166.67 | 7166.67 | $\$$ | calculus | 8.R.17 | scibench | [
{
"content": "<|im_start|>system\nA conversation between User and Assistant. The user asks a question, and the assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and<answer> </answer> tags, respectively, i.e., <think> reasoning process here </think><answer> answer here </answer>.\n<|im_end|>\n<|im_start|>user\nThe demand function for a commodity is given by\r\n$$\r\np=2000-0.1 x-0.01 x^2\r\n$$\r\nFind the consumer surplus when the sales level is 100 .\n<|im_end|>\n<|im_start|>assistant\n<think>",
"role": "user"
}
]
| logic | {
"ground_truth": {
"answer_number": "7166.67",
"unit": " $\\$$"
},
"style": "rule"
} | {
"index": 346,
"split": "train"
} |
||
The linear density in a rod $8 \mathrm{~m}$ long is $12 / \sqrt{x+1} \mathrm{~kg} / \mathrm{m}$, where $x$ is measured in meters from one end of the rod. Find the average density of the rod. | 6 | 6 | $\mathrm{~kg} / \mathrm{m}$ | calculus | 6.5.19 | scibench | [
{
"content": "<|im_start|>system\nA conversation between User and Assistant. The user asks a question, and the assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and<answer> </answer> tags, respectively, i.e., <think> reasoning process here </think><answer> answer here </answer>.\n<|im_end|>\n<|im_start|>user\nThe linear density in a rod $8 \\mathrm{~m}$ long is $12 / \\sqrt{x+1} \\mathrm{~kg} / \\mathrm{m}$, where $x$ is measured in meters from one end of the rod. Find the average density of the rod.\n<|im_end|>\n<|im_start|>assistant\n<think>",
"role": "user"
}
]
| logic | {
"ground_truth": {
"answer_number": "6",
"unit": " $\\mathrm{~kg} / \\mathrm{m}$"
},
"style": "rule"
} | {
"index": 347,
"split": "train"
} |
||
A variable force of $5 x^{-2}$ pounds moves an object along a straight line when it is $x$ feet from the origin. Calculate the work done in moving the object from $x=1 \mathrm{~ft}$ to $x=10 \mathrm{~ft}$. | 4.5 | 4.5 | $\mathrm{ft-lb}$ | calculus | 6.4.3 | scibench | [
{
"content": "<|im_start|>system\nA conversation between User and Assistant. The user asks a question, and the assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and<answer> </answer> tags, respectively, i.e., <think> reasoning process here </think><answer> answer here </answer>.\n<|im_end|>\n<|im_start|>user\nA variable force of $5 x^{-2}$ pounds moves an object along a straight line when it is $x$ feet from the origin. Calculate the work done in moving the object from $x=1 \\mathrm{~ft}$ to $x=10 \\mathrm{~ft}$.\n<|im_end|>\n<|im_start|>assistant\n<think>",
"role": "user"
}
]
| logic | {
"ground_truth": {
"answer_number": "4.5",
"unit": " $\\mathrm{ft-lb}$"
},
"style": "rule"
} | {
"index": 348,
"split": "train"
} |
||
One model for the spread of an epidemic is that the rate of spread is jointly proportional to the number of infected people and the number of uninfected people. In an isolated town of 5000 inhabitants, 160 people have a disease at the beginning of the week and 1200 have it at the end of the week. How long does it take for $80 \%$ of the population to become infected? | 15 | 15 | $\mathrm{days}$ | calculus | 9.R.19 | scibench | [
{
"content": "<|im_start|>system\nA conversation between User and Assistant. The user asks a question, and the assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and<answer> </answer> tags, respectively, i.e., <think> reasoning process here </think><answer> answer here </answer>.\n<|im_end|>\n<|im_start|>user\nOne model for the spread of an epidemic is that the rate of spread is jointly proportional to the number of infected people and the number of uninfected people. In an isolated town of 5000 inhabitants, 160 people have a disease at the beginning of the week and 1200 have it at the end of the week. How long does it take for $80 \\%$ of the population to become infected?\n<|im_end|>\n<|im_start|>assistant\n<think>",
"role": "user"
}
]
| logic | {
"ground_truth": {
"answer_number": "15",
"unit": " $\\mathrm{days}$"
},
"style": "rule"
} | {
"index": 349,
"split": "train"
} |
||
Find the volume of the described solid S. A tetrahedron with three mutually perpendicular faces and three mutually perpendicular edges with lengths $3 \mathrm{~cm}$, $4 \mathrm{~cm}$, and $5 \mathrm{~cm}$ | 10 | 10 | $\mathrm{cm}^3$ | calculus | 6.2.53 | scibench | [
{
"content": "<|im_start|>system\nA conversation between User and Assistant. The user asks a question, and the assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and<answer> </answer> tags, respectively, i.e., <think> reasoning process here </think><answer> answer here </answer>.\n<|im_end|>\n<|im_start|>user\nFind the volume of the described solid S. A tetrahedron with three mutually perpendicular faces and three mutually perpendicular edges with lengths $3 \\mathrm{~cm}$, $4 \\mathrm{~cm}$, and $5 \\mathrm{~cm}$\n<|im_end|>\n<|im_start|>assistant\n<think>",
"role": "user"
}
]
| logic | {
"ground_truth": {
"answer_number": "10",
"unit": " $\\mathrm{cm}^3$"
},
"style": "rule"
} | {
"index": 350,
"split": "train"
} |
||
The base of a solid is a circular disk with radius 3 . Find the volume of the solid if parallel cross-sections perpendicular to the base are isosceles right triangles with hypotenuse lying along the base. | 36 | 36 | calculus | 6.R.23 | review problem | scibench | [
{
"content": "<|im_start|>system\nA conversation between User and Assistant. The user asks a question, and the assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and<answer> </answer> tags, respectively, i.e., <think> reasoning process here </think><answer> answer here </answer>.\n<|im_end|>\n<|im_start|>user\nThe base of a solid is a circular disk with radius 3 . Find the volume of the solid if parallel cross-sections perpendicular to the base are isosceles right triangles with hypotenuse lying along the base.\n<|im_end|>\n<|im_start|>assistant\n<think>",
"role": "user"
}
]
| logic | {
"ground_truth": {
"answer_number": "36",
"unit": " "
},
"style": "rule"
} | {
"index": 351,
"split": "train"
} |
||
A projectile is fired with an initial speed of $200 \mathrm{~m} / \mathrm{s}$ and angle of elevation $60^{\circ}$. Find the speed at impact. | 200 | 200 | $\mathrm{m}/\mathrm{s}$ | calculus | 13.4.23(c) | scibench | [
{
"content": "<|im_start|>system\nA conversation between User and Assistant. The user asks a question, and the assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and<answer> </answer> tags, respectively, i.e., <think> reasoning process here </think><answer> answer here </answer>.\n<|im_end|>\n<|im_start|>user\nA projectile is fired with an initial speed of $200 \\mathrm{~m} / \\mathrm{s}$ and angle of elevation $60^{\\circ}$. Find the speed at impact.\n<|im_end|>\n<|im_start|>assistant\n<think>",
"role": "user"
}
]
| logic | {
"ground_truth": {
"answer_number": "200",
"unit": " $\\mathrm{m}/\\mathrm{s}$"
},
"style": "rule"
} | {
"index": 352,
"split": "train"
} |
||
A force of $30 \mathrm{~N}$ is required to maintain a spring stretched from its natural length of $12 \mathrm{~cm}$ to a length of $15 \mathrm{~cm}$. How much work is done in stretching the spring from $12 \mathrm{~cm}$ to $20 \mathrm{~cm}$ ?
| 3.2 | 3.2 | $\mathrm{J}$ | calculus | 6.R.27 | review problem | scibench | [
{
"content": "<|im_start|>system\nA conversation between User and Assistant. The user asks a question, and the assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and<answer> </answer> tags, respectively, i.e., <think> reasoning process here </think><answer> answer here </answer>.\n<|im_end|>\n<|im_start|>user\nA force of $30 \\mathrm{~N}$ is required to maintain a spring stretched from its natural length of $12 \\mathrm{~cm}$ to a length of $15 \\mathrm{~cm}$. How much work is done in stretching the spring from $12 \\mathrm{~cm}$ to $20 \\mathrm{~cm}$ ?\r\n\n<|im_end|>\n<|im_start|>assistant\n<think>",
"role": "user"
}
]
| logic | {
"ground_truth": {
"answer_number": "3.2",
"unit": " $\\mathrm{J}$"
},
"style": "rule"
} | {
"index": 353,
"split": "train"
} |
|
Use Poiseuille's Law to calculate the rate of flow in a small human artery where we can take $\eta=0.027, R=0.008 \mathrm{~cm}$, $I=2 \mathrm{~cm}$, and $P=4000$ dynes $/ \mathrm{cm}^2$. | 1.19 | 1.19 | $\mathrm{10^{-4}} \mathrm{~cm}^3/\mathrm{s}$ | calculus | 8.4.15 | scibench | [
{
"content": "<|im_start|>system\nA conversation between User and Assistant. The user asks a question, and the assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and<answer> </answer> tags, respectively, i.e., <think> reasoning process here </think><answer> answer here </answer>.\n<|im_end|>\n<|im_start|>user\nUse Poiseuille's Law to calculate the rate of flow in a small human artery where we can take $\\eta=0.027, R=0.008 \\mathrm{~cm}$, $I=2 \\mathrm{~cm}$, and $P=4000$ dynes $/ \\mathrm{cm}^2$.\n<|im_end|>\n<|im_start|>assistant\n<think>",
"role": "user"
}
]
| logic | {
"ground_truth": {
"answer_number": "1.19",
"unit": " $\\mathrm{10^{-4}} \\mathrm{~cm}^3/\\mathrm{s}$"
},
"style": "rule"
} | {
"index": 354,
"split": "train"
} |
||
If the distribution of $Y$ is $b(n, 0.25)$, give a lower bound for $P(|Y / n-0.25|<0.05)$ when $n=100$. | $0.25$ | 0.25 | stat | 5.8-5 (a) | scibench | [
{
"content": "<|im_start|>system\nA conversation between User and Assistant. The user asks a question, and the assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and<answer> </answer> tags, respectively, i.e., <think> reasoning process here </think><answer> answer here </answer>.\n<|im_end|>\n<|im_start|>user\nIf the distribution of $Y$ is $b(n, 0.25)$, give a lower bound for $P(|Y / n-0.25|<0.05)$ when $n=100$.\n<|im_end|>\n<|im_start|>assistant\n<think>",
"role": "user"
}
]
| logic | {
"ground_truth": {
"answer_number": "0.25",
"unit": " "
},
"style": "rule"
} | {
"index": 355,
"split": "train"
} |
|||
A device contains three components, each of which has a lifetime in hours with the pdf
$$
f(x)=\frac{2 x}{10^2} e^{-(x / 10)^2}, \quad 0 < x < \infty .
$$
The device fails with the failure of one of the components. Assuming independent lifetimes, what is the probability that the device fails in the first hour of its operation? HINT: $G(y)=P(Y \leq y)=1-P(Y>y)=1-P$ (all three $>y$ ). | $0.03$ | 0.03 | stat | 5.3-13 | scibench | [
{
"content": "<|im_start|>system\nA conversation between User and Assistant. The user asks a question, and the assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and<answer> </answer> tags, respectively, i.e., <think> reasoning process here </think><answer> answer here </answer>.\n<|im_end|>\n<|im_start|>user\nA device contains three components, each of which has a lifetime in hours with the pdf\r\n$$\r\nf(x)=\\frac{2 x}{10^2} e^{-(x / 10)^2}, \\quad 0 < x < \\infty .\r\n$$\r\nThe device fails with the failure of one of the components. Assuming independent lifetimes, what is the probability that the device fails in the first hour of its operation? HINT: $G(y)=P(Y \\leq y)=1-P(Y>y)=1-P$ (all three $>y$ ).\n<|im_end|>\n<|im_start|>assistant\n<think>",
"role": "user"
}
]
| logic | {
"ground_truth": {
"answer_number": "0.03",
"unit": " "
},
"style": "rule"
} | {
"index": 356,
"split": "train"
} |
|||
The tensile strength $X$ of paper, in pounds per square inch, has $\mu=30$ and $\sigma=3$. A random sample of size $n=100$ is taken from the distribution of tensile strengths. Compute the probability that the sample mean $\bar{X}$ is greater than 29.5 pounds per square inch. | $0.9522$ | 0.9522 | stat | 5.6-13 | scibench | [
{
"content": "<|im_start|>system\nA conversation between User and Assistant. The user asks a question, and the assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and<answer> </answer> tags, respectively, i.e., <think> reasoning process here </think><answer> answer here </answer>.\n<|im_end|>\n<|im_start|>user\nThe tensile strength $X$ of paper, in pounds per square inch, has $\\mu=30$ and $\\sigma=3$. A random sample of size $n=100$ is taken from the distribution of tensile strengths. Compute the probability that the sample mean $\\bar{X}$ is greater than 29.5 pounds per square inch.\n<|im_end|>\n<|im_start|>assistant\n<think>",
"role": "user"
}
]
| logic | {
"ground_truth": {
"answer_number": "0.9522",
"unit": " "
},
"style": "rule"
} | {
"index": 357,
"split": "train"
} |
|||
Let $\bar{X}$ be the mean of a random sample of size 36 from an exponential distribution with mean 3 . Approximate $P(2.5 \leq \bar{X} \leq 4)$ | $0.8185$ | 0.8185 | stat | 5.6-3 | scibench | [
{
"content": "<|im_start|>system\nA conversation between User and Assistant. The user asks a question, and the assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and<answer> </answer> tags, respectively, i.e., <think> reasoning process here </think><answer> answer here </answer>.\n<|im_end|>\n<|im_start|>user\nLet $\\bar{X}$ be the mean of a random sample of size 36 from an exponential distribution with mean 3 . Approximate $P(2.5 \\leq \\bar{X} \\leq 4)$\n<|im_end|>\n<|im_start|>assistant\n<think>",
"role": "user"
}
]
| logic | {
"ground_truth": {
"answer_number": "0.8185",
"unit": " "
},
"style": "rule"
} | {
"index": 358,
"split": "train"
} |
|||
Let $X_1, X_2$ be a random sample of size $n=2$ from a distribution with pdf $f(x)=3 x^2, 0 < x < 1$. Determine $P\left(\max X_i < 3 / 4\right)=P\left(X_1<3 / 4, X_2<3 / 4\right)$ | $\frac{729}{4096}$ | 0.178 | stat | 5.3-9 | scibench | [
{
"content": "<|im_start|>system\nA conversation between User and Assistant. The user asks a question, and the assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and<answer> </answer> tags, respectively, i.e., <think> reasoning process here </think><answer> answer here </answer>.\n<|im_end|>\n<|im_start|>user\nLet $X_1, X_2$ be a random sample of size $n=2$ from a distribution with pdf $f(x)=3 x^2, 0 < x < 1$. Determine $P\\left(\\max X_i < 3 / 4\\right)=P\\left(X_1<3 / 4, X_2<3 / 4\\right)$\n<|im_end|>\n<|im_start|>assistant\n<think>",
"role": "user"
}
]
| logic | {
"ground_truth": {
"answer_number": "0.178",
"unit": " "
},
"style": "rule"
} | {
"index": 359,
"split": "train"
} |
|||
Let $X$ equal the tarsus length for a male grackle. Assume that the distribution of $X$ is $N(\mu, 4.84)$. Find the sample size $n$ that is needed so that we are $95 \%$ confident that the maximum error of the estimate of $\mu$ is 0.4 . | $117$ | 117 | stat | 7.4-1 | scibench | [
{
"content": "<|im_start|>system\nA conversation between User and Assistant. The user asks a question, and the assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and<answer> </answer> tags, respectively, i.e., <think> reasoning process here </think><answer> answer here </answer>.\n<|im_end|>\n<|im_start|>user\nLet $X$ equal the tarsus length for a male grackle. Assume that the distribution of $X$ is $N(\\mu, 4.84)$. Find the sample size $n$ that is needed so that we are $95 \\%$ confident that the maximum error of the estimate of $\\mu$ is 0.4 .\n<|im_end|>\n<|im_start|>assistant\n<think>",
"role": "user"
}
]
| logic | {
"ground_truth": {
"answer_number": "117",
"unit": " "
},
"style": "rule"
} | {
"index": 360,
"split": "train"
} |
|||
In a study concerning a new treatment of a certain disease, two groups of 25 participants in each were followed for five years. Those in one group took the old treatment and those in the other took the new treatment. The theoretical dropout rate for an individual was $50 \%$ in both groups over that 5 -year period. Let $X$ be the number that dropped out in the first group and $Y$ the number in the second group. Assuming independence where needed, give the sum that equals the probability that $Y \geq X+2$. HINT: What is the distribution of $Y-X+25$ ? | $0.3359$ | 0.3359 | stat | 5.4-17 | scibench | [
{
"content": "<|im_start|>system\nA conversation between User and Assistant. The user asks a question, and the assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and<answer> </answer> tags, respectively, i.e., <think> reasoning process here </think><answer> answer here </answer>.\n<|im_end|>\n<|im_start|>user\nIn a study concerning a new treatment of a certain disease, two groups of 25 participants in each were followed for five years. Those in one group took the old treatment and those in the other took the new treatment. The theoretical dropout rate for an individual was $50 \\%$ in both groups over that 5 -year period. Let $X$ be the number that dropped out in the first group and $Y$ the number in the second group. Assuming independence where needed, give the sum that equals the probability that $Y \\geq X+2$. HINT: What is the distribution of $Y-X+25$ ?\n<|im_end|>\n<|im_start|>assistant\n<think>",
"role": "user"
}
]
| logic | {
"ground_truth": {
"answer_number": "0.3359",
"unit": " "
},
"style": "rule"
} | {
"index": 361,
"split": "train"
} |
|||
Let $X$ and $Y$ have a bivariate normal distribution with correlation coefficient $\rho$. To test $H_0: \rho=0$ against $H_1: \rho \neq 0$, a random sample of $n$ pairs of observations is selected. Suppose that the sample correlation coefficient is $r=0.68$. Using a significance level of $\alpha=0.05$, find the smallest value of the sample size $n$ so that $H_0$ is rejected. | $9$ | 9 | stat | 9.6-11 | scibench | [
{
"content": "<|im_start|>system\nA conversation between User and Assistant. The user asks a question, and the assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and<answer> </answer> tags, respectively, i.e., <think> reasoning process here </think><answer> answer here </answer>.\n<|im_end|>\n<|im_start|>user\nLet $X$ and $Y$ have a bivariate normal distribution with correlation coefficient $\\rho$. To test $H_0: \\rho=0$ against $H_1: \\rho \\neq 0$, a random sample of $n$ pairs of observations is selected. Suppose that the sample correlation coefficient is $r=0.68$. Using a significance level of $\\alpha=0.05$, find the smallest value of the sample size $n$ so that $H_0$ is rejected.\n<|im_end|>\n<|im_start|>assistant\n<think>",
"role": "user"
}
]
| logic | {
"ground_truth": {
"answer_number": "9",
"unit": " "
},
"style": "rule"
} | {
"index": 362,
"split": "train"
} |
|||
In order to estimate the proportion, $p$, of a large class of college freshmen that had high school GPAs from 3.2 to 3.6 , inclusive, a sample of $n=50$ students was taken. It was found that $y=9$ students fell into this interval. Give a point estimate of $p$. | $0.1800$ | 0.1800 | stat | 7.3-5 | scibench | [
{
"content": "<|im_start|>system\nA conversation between User and Assistant. The user asks a question, and the assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and<answer> </answer> tags, respectively, i.e., <think> reasoning process here </think><answer> answer here </answer>.\n<|im_end|>\n<|im_start|>user\nIn order to estimate the proportion, $p$, of a large class of college freshmen that had high school GPAs from 3.2 to 3.6 , inclusive, a sample of $n=50$ students was taken. It was found that $y=9$ students fell into this interval. Give a point estimate of $p$.\n<|im_end|>\n<|im_start|>assistant\n<think>",
"role": "user"
}
]
| logic | {
"ground_truth": {
"answer_number": "0.1800",
"unit": " "
},
"style": "rule"
} | {
"index": 363,
"split": "train"
} |
|||
If $\bar{X}$ and $\bar{Y}$ are the respective means of two independent random samples of the same size $n$, find $n$ if we want $\bar{x}-\bar{y} \pm 4$ to be a $90 \%$ confidence interval for $\mu_X-\mu_Y$. Assume that the standard deviations are known to be $\sigma_X=15$ and $\sigma_Y=25$. | $144$ | 144 | stat | 7.4-15 | scibench | [
{
"content": "<|im_start|>system\nA conversation between User and Assistant. The user asks a question, and the assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and<answer> </answer> tags, respectively, i.e., <think> reasoning process here </think><answer> answer here </answer>.\n<|im_end|>\n<|im_start|>user\nIf $\\bar{X}$ and $\\bar{Y}$ are the respective means of two independent random samples of the same size $n$, find $n$ if we want $\\bar{x}-\\bar{y} \\pm 4$ to be a $90 \\%$ confidence interval for $\\mu_X-\\mu_Y$. Assume that the standard deviations are known to be $\\sigma_X=15$ and $\\sigma_Y=25$.\n<|im_end|>\n<|im_start|>assistant\n<think>",
"role": "user"
}
]
| logic | {
"ground_truth": {
"answer_number": "144",
"unit": " "
},
"style": "rule"
} | {
"index": 364,
"split": "train"
} |
|||
For a public opinion poll for a close presidential election, let $p$ denote the proportion of voters who favor candidate $A$. How large a sample should be taken if we want the maximum error of the estimate of $p$ to be equal to 0.03 with $95 \%$ confidence? | $1068$ | 1068 | stat | 7.4-7 | scibench | [
{
"content": "<|im_start|>system\nA conversation between User and Assistant. The user asks a question, and the assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and<answer> </answer> tags, respectively, i.e., <think> reasoning process here </think><answer> answer here </answer>.\n<|im_end|>\n<|im_start|>user\nFor a public opinion poll for a close presidential election, let $p$ denote the proportion of voters who favor candidate $A$. How large a sample should be taken if we want the maximum error of the estimate of $p$ to be equal to 0.03 with $95 \\%$ confidence?\n<|im_end|>\n<|im_start|>assistant\n<think>",
"role": "user"
}
]
| logic | {
"ground_truth": {
"answer_number": "1068",
"unit": " "
},
"style": "rule"
} | {
"index": 365,
"split": "train"
} |
|||
Let the distribution of $T$ be $t(17)$. Find $t_{0.01}(17)$. | $2.567$ | 2.567 | stat | 5.5-15 (a) | scibench | [
{
"content": "<|im_start|>system\nA conversation between User and Assistant. The user asks a question, and the assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and<answer> </answer> tags, respectively, i.e., <think> reasoning process here </think><answer> answer here </answer>.\n<|im_end|>\n<|im_start|>user\nLet the distribution of $T$ be $t(17)$. Find $t_{0.01}(17)$.\n<|im_end|>\n<|im_start|>assistant\n<think>",
"role": "user"
}
]
| logic | {
"ground_truth": {
"answer_number": "2.567",
"unit": " "
},
"style": "rule"
} | {
"index": 366,
"split": "train"
} |
|||
Let $X_1, X_2, \ldots, X_{16}$ be a random sample from a normal distribution $N(77,25)$. Compute $P(77<\bar{X}<79.5)$. | $0.4772$ | 0.4772 | stat | 5.5-1 | scibench | [
{
"content": "<|im_start|>system\nA conversation between User and Assistant. The user asks a question, and the assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and<answer> </answer> tags, respectively, i.e., <think> reasoning process here </think><answer> answer here </answer>.\n<|im_end|>\n<|im_start|>user\nLet $X_1, X_2, \\ldots, X_{16}$ be a random sample from a normal distribution $N(77,25)$. Compute $P(77<\\bar{X}<79.5)$.\n<|im_end|>\n<|im_start|>assistant\n<think>",
"role": "user"
}
]
| logic | {
"ground_truth": {
"answer_number": "0.4772",
"unit": " "
},
"style": "rule"
} | {
"index": 367,
"split": "train"
} |
|||
5.4-19. A doorman at a hotel is trying to get three taxicabs for three different couples. The arrival of empty cabs has an exponential distribution with mean 2 minutes. Assuming independence, what is the probability that the doorman will get all three couples taken care of within 6 minutes?
| $0.5768$ | 0.5768 | stat | 5.4-19 | scibench | [
{
"content": "<|im_start|>system\nA conversation between User and Assistant. The user asks a question, and the assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and<answer> </answer> tags, respectively, i.e., <think> reasoning process here </think><answer> answer here </answer>.\n<|im_end|>\n<|im_start|>user\n5.4-19. A doorman at a hotel is trying to get three taxicabs for three different couples. The arrival of empty cabs has an exponential distribution with mean 2 minutes. Assuming independence, what is the probability that the doorman will get all three couples taken care of within 6 minutes?\r\n\n<|im_end|>\n<|im_start|>assistant\n<think>",
"role": "user"
}
]
| logic | {
"ground_truth": {
"answer_number": "0.5768",
"unit": " "
},
"style": "rule"
} | {
"index": 368,
"split": "train"
} |
|||
Consider the following two groups of women: Group 1 consists of women who spend less than $\$ 500$ annually on clothes; Group 2 comprises women who spend over $\$ 1000$ annually on clothes. Let $p_1$ and $p_2$ equal the proportions of women in these two groups, respectively, who believe that clothes are too expensive. If 1009 out of a random sample of 1230 women from group 1 and 207 out of a random sample 340 from group 2 believe that clothes are too expensive, Give a point estimate of $p_1-p_2$. | $0.2115$ | 0.2115 | stat | 7.3-9 | scibench | [
{
"content": "<|im_start|>system\nA conversation between User and Assistant. The user asks a question, and the assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and<answer> </answer> tags, respectively, i.e., <think> reasoning process here </think><answer> answer here </answer>.\n<|im_end|>\n<|im_start|>user\nConsider the following two groups of women: Group 1 consists of women who spend less than $\\$ 500$ annually on clothes; Group 2 comprises women who spend over $\\$ 1000$ annually on clothes. Let $p_1$ and $p_2$ equal the proportions of women in these two groups, respectively, who believe that clothes are too expensive. If 1009 out of a random sample of 1230 women from group 1 and 207 out of a random sample 340 from group 2 believe that clothes are too expensive, Give a point estimate of $p_1-p_2$.\n<|im_end|>\n<|im_start|>assistant\n<think>",
"role": "user"
}
]
| logic | {
"ground_truth": {
"answer_number": "0.2115",
"unit": " "
},
"style": "rule"
} | {
"index": 369,
"split": "train"
} |
|||
Given below example: Approximate $P(39.75 \leq \bar{X} \leq 41.25)$, where $\bar{X}$ is the mean of a random sample of size 32 from a distribution with mean $\mu=40$ and variance $\sigma^2=8$. In the above example, compute $P(1.7 \leq Y \leq 3.2)$ with $n=4$ | $0.6749$ | 0.6749 | stat | 5.6-9 | scibench | [
{
"content": "<|im_start|>system\nA conversation between User and Assistant. The user asks a question, and the assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and<answer> </answer> tags, respectively, i.e., <think> reasoning process here </think><answer> answer here </answer>.\n<|im_end|>\n<|im_start|>user\nGiven below example: Approximate $P(39.75 \\leq \\bar{X} \\leq 41.25)$, where $\\bar{X}$ is the mean of a random sample of size 32 from a distribution with mean $\\mu=40$ and variance $\\sigma^2=8$. In the above example, compute $P(1.7 \\leq Y \\leq 3.2)$ with $n=4$\n<|im_end|>\n<|im_start|>assistant\n<think>",
"role": "user"
}
]
| logic | {
"ground_truth": {
"answer_number": "0.6749",
"unit": " "
},
"style": "rule"
} | {
"index": 370,
"split": "train"
} |
|||
If the distribution of $Y$ is $b(n, 0.25)$, give a lower bound for $P(|Y / n-0.25|<0.05)$ when $n=1000$. | $0.925$ | 0.925 | stat | 5.8-5 | scibench | [
{
"content": "<|im_start|>system\nA conversation between User and Assistant. The user asks a question, and the assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and<answer> </answer> tags, respectively, i.e., <think> reasoning process here </think><answer> answer here </answer>.\n<|im_end|>\n<|im_start|>user\nIf the distribution of $Y$ is $b(n, 0.25)$, give a lower bound for $P(|Y / n-0.25|<0.05)$ when $n=1000$.\n<|im_end|>\n<|im_start|>assistant\n<think>",
"role": "user"
}
]
| logic | {
"ground_truth": {
"answer_number": "0.925",
"unit": " "
},
"style": "rule"
} | {
"index": 371,
"split": "train"
} |
|||
Let $Y_1 < Y_2 < Y_3 < Y_4 < Y_5 < Y_6$ be the order statistics of a random sample of size $n=6$ from a distribution of the continuous type having $(100 p)$ th percentile $\pi_p$. Compute $P\left(Y_2 < \pi_{0.5} < Y_5\right)$. | $0.7812$ | 0.7812 | stat | 7.5-1 | scibench | [
{
"content": "<|im_start|>system\nA conversation between User and Assistant. The user asks a question, and the assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and<answer> </answer> tags, respectively, i.e., <think> reasoning process here </think><answer> answer here </answer>.\n<|im_end|>\n<|im_start|>user\nLet $Y_1 < Y_2 < Y_3 < Y_4 < Y_5 < Y_6$ be the order statistics of a random sample of size $n=6$ from a distribution of the continuous type having $(100 p)$ th percentile $\\pi_p$. Compute $P\\left(Y_2 < \\pi_{0.5} < Y_5\\right)$.\n<|im_end|>\n<|im_start|>assistant\n<think>",
"role": "user"
}
]
| logic | {
"ground_truth": {
"answer_number": "0.7812",
"unit": " "
},
"style": "rule"
} | {
"index": 372,
"split": "train"
} |
|||
Let $X_1, X_2$ be independent random variables representing lifetimes (in hours) of two key components of a
device that fails when and only when both components fail. Say each $X_i$ has an exponential distribution with mean 1000. Let $Y_1=\min \left(X_1, X_2\right)$ and $Y_2=\max \left(X_1, X_2\right)$, so that the space of $Y_1, Y_2$ is $ 0< y_1 < y_2 < \infty $ Find $G\left(y_1, y_2\right)=P\left(Y_1 \leq y_1, Y_2 \leq y_2\right)$. | 0.5117 | 0.5117 | stat | 5.2-13 | scibench | [
{
"content": "<|im_start|>system\nA conversation between User and Assistant. The user asks a question, and the assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and<answer> </answer> tags, respectively, i.e., <think> reasoning process here </think><answer> answer here </answer>.\n<|im_end|>\n<|im_start|>user\nLet $X_1, X_2$ be independent random variables representing lifetimes (in hours) of two key components of a\r\ndevice that fails when and only when both components fail. Say each $X_i$ has an exponential distribution with mean 1000. Let $Y_1=\\min \\left(X_1, X_2\\right)$ and $Y_2=\\max \\left(X_1, X_2\\right)$, so that the space of $Y_1, Y_2$ is $ 0< y_1 < y_2 < \\infty $ Find $G\\left(y_1, y_2\\right)=P\\left(Y_1 \\leq y_1, Y_2 \\leq y_2\\right)$.\n<|im_end|>\n<|im_start|>assistant\n<think>",
"role": "user"
}
]
| logic | {
"ground_truth": {
"answer_number": "0.5117",
"unit": " "
},
"style": "rule"
} | {
"index": 373,
"split": "train"
} |
|||
Let $Z_1, Z_2, \ldots, Z_7$ be a random sample from the standard normal distribution $N(0,1)$. Let $W=Z_1^2+Z_2^2+$ $\cdots+Z_7^2$. Find $P(1.69 < W < 14.07)$ | $0.925$ | 0.925 | stat | 5.4-5 | scibench | [
{
"content": "<|im_start|>system\nA conversation between User and Assistant. The user asks a question, and the assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and<answer> </answer> tags, respectively, i.e., <think> reasoning process here </think><answer> answer here </answer>.\n<|im_end|>\n<|im_start|>user\nLet $Z_1, Z_2, \\ldots, Z_7$ be a random sample from the standard normal distribution $N(0,1)$. Let $W=Z_1^2+Z_2^2+$ $\\cdots+Z_7^2$. Find $P(1.69 < W < 14.07)$\n<|im_end|>\n<|im_start|>assistant\n<think>",
"role": "user"
}
]
| logic | {
"ground_truth": {
"answer_number": "0.925",
"unit": " "
},
"style": "rule"
} | {
"index": 374,
"split": "train"
} |
|||
Let $X_1$ and $X_2$ be independent Poisson random variables with respective means $\lambda_1=2$ and $\lambda_2=3$. Find $P\left(X_1=3, X_2=5\right)$. HINT. Note that this event can occur if and only if $\left\{X_1=1, X_2=0\right\}$ or $\left\{X_1=0, X_2=1\right\}$. | 0.0182 | 0.0182 | stat | 5.3-1 | scibench | [
{
"content": "<|im_start|>system\nA conversation between User and Assistant. The user asks a question, and the assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and<answer> </answer> tags, respectively, i.e., <think> reasoning process here </think><answer> answer here </answer>.\n<|im_end|>\n<|im_start|>user\nLet $X_1$ and $X_2$ be independent Poisson random variables with respective means $\\lambda_1=2$ and $\\lambda_2=3$. Find $P\\left(X_1=3, X_2=5\\right)$. HINT. Note that this event can occur if and only if $\\left\\{X_1=1, X_2=0\\right\\}$ or $\\left\\{X_1=0, X_2=1\\right\\}$.\n<|im_end|>\n<|im_start|>assistant\n<think>",
"role": "user"
}
]
| logic | {
"ground_truth": {
"answer_number": "0.0182",
"unit": " "
},
"style": "rule"
} | {
"index": 375,
"split": "train"
} |
|||
Let $Y$ be the number of defectives in a box of 50 articles taken from the output of a machine. Each article is defective with probability 0.01 . Find the probability that $Y=0,1,2$, or 3 By using the binomial distribution. | $0.9984$ | 0.9984 | stat | 5.9-1 (a) | scibench | [
{
"content": "<|im_start|>system\nA conversation between User and Assistant. The user asks a question, and the assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and<answer> </answer> tags, respectively, i.e., <think> reasoning process here </think><answer> answer here </answer>.\n<|im_end|>\n<|im_start|>user\nLet $Y$ be the number of defectives in a box of 50 articles taken from the output of a machine. Each article is defective with probability 0.01 . Find the probability that $Y=0,1,2$, or 3 By using the binomial distribution.\n<|im_end|>\n<|im_start|>assistant\n<think>",
"role": "user"
}
]
| logic | {
"ground_truth": {
"answer_number": "0.9984",
"unit": " "
},
"style": "rule"
} | {
"index": 376,
"split": "train"
} |
|||
Some dentists were interested in studying the fusion of embryonic rat palates by a standard transplantation technique. When no treatment is used, the probability of fusion equals approximately 0.89 . The dentists would like to estimate $p$, the probability of fusion, when vitamin A is lacking. How large a sample $n$ of rat embryos is needed for $y / n \pm 0.10$ to be a $95 \%$ confidence interval for $p$ ? | $38$ | 38 | stat | 7.4-11 | scibench | [
{
"content": "<|im_start|>system\nA conversation between User and Assistant. The user asks a question, and the assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and<answer> </answer> tags, respectively, i.e., <think> reasoning process here </think><answer> answer here </answer>.\n<|im_end|>\n<|im_start|>user\nSome dentists were interested in studying the fusion of embryonic rat palates by a standard transplantation technique. When no treatment is used, the probability of fusion equals approximately 0.89 . The dentists would like to estimate $p$, the probability of fusion, when vitamin A is lacking. How large a sample $n$ of rat embryos is needed for $y / n \\pm 0.10$ to be a $95 \\%$ confidence interval for $p$ ?\n<|im_end|>\n<|im_start|>assistant\n<think>",
"role": "user"
}
]
| logic | {
"ground_truth": {
"answer_number": "38",
"unit": " "
},
"style": "rule"
} | {
"index": 377,
"split": "train"
} |
|||
To determine the effect of $100 \%$ nitrate on the growth of pea plants, several specimens were planted and then watered with $100 \%$ nitrate every day. At the end of
two weeks, the plants were measured. Here are data on seven of them:
$$
\begin{array}{lllllll}
17.5 & 14.5 & 15.2 & 14.0 & 17.3 & 18.0 & 13.8
\end{array}
$$
Assume that these data are a random sample from a normal distribution $N\left(\mu, \sigma^2\right)$. Find the value of a point estimate of $\mu$. | $15.757$ | 15.757 | stat | 7.1-3 | scibench | [
{
"content": "<|im_start|>system\nA conversation between User and Assistant. The user asks a question, and the assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and<answer> </answer> tags, respectively, i.e., <think> reasoning process here </think><answer> answer here </answer>.\n<|im_end|>\n<|im_start|>user\nTo determine the effect of $100 \\%$ nitrate on the growth of pea plants, several specimens were planted and then watered with $100 \\%$ nitrate every day. At the end of\r\ntwo weeks, the plants were measured. Here are data on seven of them:\r\n$$\r\n\\begin{array}{lllllll}\r\n17.5 & 14.5 & 15.2 & 14.0 & 17.3 & 18.0 & 13.8\r\n\\end{array}\r\n$$\r\nAssume that these data are a random sample from a normal distribution $N\\left(\\mu, \\sigma^2\\right)$. Find the value of a point estimate of $\\mu$.\n<|im_end|>\n<|im_start|>assistant\n<think>",
"role": "user"
}
]
| logic | {
"ground_truth": {
"answer_number": "15.757",
"unit": " "
},
"style": "rule"
} | {
"index": 378,
"split": "train"
} |
|||
Suppose that the distribution of the weight of a prepackaged '1-pound bag' of carrots is $N\left(1.18,0.07^2\right)$ and the distribution of the weight of a prepackaged '3-pound bag' of carrots is $N\left(3.22,0.09^2\right)$. Selecting bags at random, find the probability that the sum of three 1-pound bags exceeds the weight of one 3-pound bag. HInT: First determine the distribution of $Y$, the sum of the three, and then compute $P(Y>W)$, where $W$ is the weight of the 3-pound bag. | $0.9830$ | 0.9830 | stat | 5.5-7 | scibench | [
{
"content": "<|im_start|>system\nA conversation between User and Assistant. The user asks a question, and the assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and<answer> </answer> tags, respectively, i.e., <think> reasoning process here </think><answer> answer here </answer>.\n<|im_end|>\n<|im_start|>user\nSuppose that the distribution of the weight of a prepackaged '1-pound bag' of carrots is $N\\left(1.18,0.07^2\\right)$ and the distribution of the weight of a prepackaged '3-pound bag' of carrots is $N\\left(3.22,0.09^2\\right)$. Selecting bags at random, find the probability that the sum of three 1-pound bags exceeds the weight of one 3-pound bag. HInT: First determine the distribution of $Y$, the sum of the three, and then compute $P(Y>W)$, where $W$ is the weight of the 3-pound bag.\n<|im_end|>\n<|im_start|>assistant\n<think>",
"role": "user"
}
]
| logic | {
"ground_truth": {
"answer_number": "0.9830",
"unit": " "
},
"style": "rule"
} | {
"index": 379,
"split": "train"
} |
|||
The distributions of incomes in two cities follow the two Pareto-type pdfs $$ f(x)=\frac{2}{x^3}, 1 < x < \infty , \text { and } g(y)= \frac{3}{y^4} , \quad 1 < y < \infty,$$ respectively. Here one unit represents $ 20,000$. One person with income is selected at random from each city. Let $X$ and $Y$ be their respective incomes. Compute $P(X < Y)$. | $\frac{2}{5}$ | 0.4 | stat | 5.3-7 | scibench | [
{
"content": "<|im_start|>system\nA conversation between User and Assistant. The user asks a question, and the assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and<answer> </answer> tags, respectively, i.e., <think> reasoning process here </think><answer> answer here </answer>.\n<|im_end|>\n<|im_start|>user\nThe distributions of incomes in two cities follow the two Pareto-type pdfs $$ f(x)=\\frac{2}{x^3}, 1 < x < \\infty , \\text { and } g(y)= \\frac{3}{y^4} , \\quad 1 < y < \\infty,$$ respectively. Here one unit represents $ 20,000$. One person with income is selected at random from each city. Let $X$ and $Y$ be their respective incomes. Compute $P(X < Y)$.\n<|im_end|>\n<|im_start|>assistant\n<think>",
"role": "user"
}
]
| logic | {
"ground_truth": {
"answer_number": "0.4",
"unit": " "
},
"style": "rule"
} | {
"index": 380,
"split": "train"
} |
|||
Let $p$ equal the proportion of triathletes who suffered a training-related overuse injury during the past year. Out of 330 triathletes who responded to a survey, 167 indicated that they had suffered such an injury during the past year. Use these data to give a point estimate of $p$. | $0.5061$ | 0.5061 | stat | 7.3-3 | scibench | [
{
"content": "<|im_start|>system\nA conversation between User and Assistant. The user asks a question, and the assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and<answer> </answer> tags, respectively, i.e., <think> reasoning process here </think><answer> answer here </answer>.\n<|im_end|>\n<|im_start|>user\nLet $p$ equal the proportion of triathletes who suffered a training-related overuse injury during the past year. Out of 330 triathletes who responded to a survey, 167 indicated that they had suffered such an injury during the past year. Use these data to give a point estimate of $p$.\n<|im_end|>\n<|im_start|>assistant\n<think>",
"role": "user"
}
]
| logic | {
"ground_truth": {
"answer_number": "0.5061",
"unit": " "
},
"style": "rule"
} | {
"index": 381,
"split": "train"
} |
|||
One characteristic of a car's storage console that is checked by the manufacturer is the time in seconds that it takes for the lower storage compartment door to open completely. A random sample of size $n=5$ yielded the following times:
$\begin{array}{lllll}1.1 & 0.9 & 1.4 & 1.1 & 1.0\end{array}$ Find the sample mean, $\bar{x}$. | $1.1$ | 1.1 | stat | 6.1-1 | scibench | [
{
"content": "<|im_start|>system\nA conversation between User and Assistant. The user asks a question, and the assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and<answer> </answer> tags, respectively, i.e., <think> reasoning process here </think><answer> answer here </answer>.\n<|im_end|>\n<|im_start|>user\nOne characteristic of a car's storage console that is checked by the manufacturer is the time in seconds that it takes for the lower storage compartment door to open completely. A random sample of size $n=5$ yielded the following times:\r\n$\\begin{array}{lllll}1.1 & 0.9 & 1.4 & 1.1 & 1.0\\end{array}$ Find the sample mean, $\\bar{x}$.\n<|im_end|>\n<|im_start|>assistant\n<think>",
"role": "user"
}
]
| logic | {
"ground_truth": {
"answer_number": "1.1",
"unit": " "
},
"style": "rule"
} | {
"index": 382,
"split": "train"
} |
|||
Let $X_1, X_2, \ldots, X_{16}$ be a random sample from a normal distribution $N(77,25)$. Compute $P(74.2<\bar{X}<78.4)$. | $0.8561$ | 0.8561 | stat | 5.5-1 (b) | scibench | [
{
"content": "<|im_start|>system\nA conversation between User and Assistant. The user asks a question, and the assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and<answer> </answer> tags, respectively, i.e., <think> reasoning process here </think><answer> answer here </answer>.\n<|im_end|>\n<|im_start|>user\nLet $X_1, X_2, \\ldots, X_{16}$ be a random sample from a normal distribution $N(77,25)$. Compute $P(74.2<\\bar{X}<78.4)$.\n<|im_end|>\n<|im_start|>assistant\n<think>",
"role": "user"
}
]
| logic | {
"ground_truth": {
"answer_number": "0.8561",
"unit": " "
},
"style": "rule"
} | {
"index": 383,
"split": "train"
} |
|||
Let $X_1$ and $X_2$ be independent random variables with probability density functions $f_1\left(x_1\right)=2 x_1, 0 < x_1 <1 $, and $f_2 \left(x_2\right) = 4x_2^3$ , $0 < x_2 < 1 $, respectively. Compute $P \left(0.5 < X_1 < 1\right.$ and $\left.0.4 < X_2 < 0.8\right)$. | $\frac{36}{125}$
| 1.44 | stat | 5.3-3 | scibench | [
{
"content": "<|im_start|>system\nA conversation between User and Assistant. The user asks a question, and the assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and<answer> </answer> tags, respectively, i.e., <think> reasoning process here </think><answer> answer here </answer>.\n<|im_end|>\n<|im_start|>user\nLet $X_1$ and $X_2$ be independent random variables with probability density functions $f_1\\left(x_1\\right)=2 x_1, 0 < x_1 <1 $, and $f_2 \\left(x_2\\right) = 4x_2^3$ , $0 < x_2 < 1 $, respectively. Compute $P \\left(0.5 < X_1 < 1\\right.$ and $\\left.0.4 < X_2 < 0.8\\right)$.\n<|im_end|>\n<|im_start|>assistant\n<think>",
"role": "user"
}
]
| logic | {
"ground_truth": {
"answer_number": "1.44",
"unit": " "
},
"style": "rule"
} | {
"index": 384,
"split": "train"
} |
|||
If $X$ is a random variable with mean 33 and variance 16, use Chebyshev's inequality to find A lower bound for $P(23 < X < 43)$. | $0.84$ | 0.84 | stat | 5.8-1 (a) | scibench | [
{
"content": "<|im_start|>system\nA conversation between User and Assistant. The user asks a question, and the assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and<answer> </answer> tags, respectively, i.e., <think> reasoning process here </think><answer> answer here </answer>.\n<|im_end|>\n<|im_start|>user\nIf $X$ is a random variable with mean 33 and variance 16, use Chebyshev's inequality to find A lower bound for $P(23 < X < 43)$.\n<|im_end|>\n<|im_start|>assistant\n<think>",
"role": "user"
}
]
| logic | {
"ground_truth": {
"answer_number": "0.84",
"unit": " "
},
"style": "rule"
} | {
"index": 385,
"split": "train"
} |
|||
Let $Y_1 < Y_2 < \cdots < Y_8$ be the order statistics of eight independent observations from a continuous-type distribution with 70 th percentile $\pi_{0.7}=27.3$. Determine $P\left(Y_7<27.3\right)$. | $0.2553$ | 0.2553 | stat | 6.3-5 | scibench | [
{
"content": "<|im_start|>system\nA conversation between User and Assistant. The user asks a question, and the assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and<answer> </answer> tags, respectively, i.e., <think> reasoning process here </think><answer> answer here </answer>.\n<|im_end|>\n<|im_start|>user\nLet $Y_1 < Y_2 < \\cdots < Y_8$ be the order statistics of eight independent observations from a continuous-type distribution with 70 th percentile $\\pi_{0.7}=27.3$. Determine $P\\left(Y_7<27.3\\right)$.\n<|im_end|>\n<|im_start|>assistant\n<think>",
"role": "user"
}
]
| logic | {
"ground_truth": {
"answer_number": "0.2553",
"unit": " "
},
"style": "rule"
} | {
"index": 386,
"split": "train"
} |
|||
Let $X$ and $Y$ be independent with distributions $N(5,16)$ and $N(6,9)$, respectively. Evaluate $P(X>Y)=$ $P(X-Y>0)$. | $0.4207$ | 0.4207 | stat | 5.4-21 | scibench | [
{
"content": "<|im_start|>system\nA conversation between User and Assistant. The user asks a question, and the assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and<answer> </answer> tags, respectively, i.e., <think> reasoning process here </think><answer> answer here </answer>.\n<|im_end|>\n<|im_start|>user\nLet $X$ and $Y$ be independent with distributions $N(5,16)$ and $N(6,9)$, respectively. Evaluate $P(X>Y)=$ $P(X-Y>0)$.\n<|im_end|>\n<|im_start|>assistant\n<think>",
"role": "user"
}
]
| logic | {
"ground_truth": {
"answer_number": "0.4207",
"unit": " "
},
"style": "rule"
} | {
"index": 387,
"split": "train"
} |
|||
A quality engineer wanted to be $98 \%$ confident that the maximum error of the estimate of the mean strength, $\mu$, of the left hinge on a vanity cover molded by a machine is 0.25 . A preliminary sample of size $n=32$ parts yielded a sample mean of $\bar{x}=35.68$ and a standard deviation of $s=1.723$. How large a sample is required? | $257$ | 257 | stat | 7.4-5 | scibench | [
{
"content": "<|im_start|>system\nA conversation between User and Assistant. The user asks a question, and the assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and<answer> </answer> tags, respectively, i.e., <think> reasoning process here </think><answer> answer here </answer>.\n<|im_end|>\n<|im_start|>user\nA quality engineer wanted to be $98 \\%$ confident that the maximum error of the estimate of the mean strength, $\\mu$, of the left hinge on a vanity cover molded by a machine is 0.25 . A preliminary sample of size $n=32$ parts yielded a sample mean of $\\bar{x}=35.68$ and a standard deviation of $s=1.723$. How large a sample is required?\n<|im_end|>\n<|im_start|>assistant\n<think>",
"role": "user"
}
]
| logic | {
"ground_truth": {
"answer_number": "257",
"unit": " "
},
"style": "rule"
} | {
"index": 388,
"split": "train"
} |
|||
Let the distribution of $W$ be $F(8,4)$. Find the following: $F_{0.01}(8,4)$. | 14.80 | 14.80 | stat | 5.2-5 | scibench | [
{
"content": "<|im_start|>system\nA conversation between User and Assistant. The user asks a question, and the assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and<answer> </answer> tags, respectively, i.e., <think> reasoning process here </think><answer> answer here </answer>.\n<|im_end|>\n<|im_start|>user\nLet the distribution of $W$ be $F(8,4)$. Find the following: $F_{0.01}(8,4)$.\n<|im_end|>\n<|im_start|>assistant\n<think>",
"role": "user"
}
]
| logic | {
"ground_truth": {
"answer_number": "14.80",
"unit": " "
},
"style": "rule"
} | {
"index": 389,
"split": "train"
} |
|||
If the distribution of $Y$ is $b(n, 0.25)$, give a lower bound for $P(|Y / n-0.25|<0.05)$ when $n=500$. | $0.85$ | 0.85 | stat | 5.8-5 | scibench | [
{
"content": "<|im_start|>system\nA conversation between User and Assistant. The user asks a question, and the assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and<answer> </answer> tags, respectively, i.e., <think> reasoning process here </think><answer> answer here </answer>.\n<|im_end|>\n<|im_start|>user\nIf the distribution of $Y$ is $b(n, 0.25)$, give a lower bound for $P(|Y / n-0.25|<0.05)$ when $n=500$.\n<|im_end|>\n<|im_start|>assistant\n<think>",
"role": "user"
}
]
| logic | {
"ground_truth": {
"answer_number": "0.85",
"unit": " "
},
"style": "rule"
} | {
"index": 390,
"split": "train"
} |
|||
Let $\bar{X}$ be the mean of a random sample of size 12 from the uniform distribution on the interval $(0,1)$. Approximate $P(1 / 2 \leq \bar{X} \leq 2 / 3)$. | $0.4772$ | 0.4772 | stat | 5.6-1 | scibench | [
{
"content": "<|im_start|>system\nA conversation between User and Assistant. The user asks a question, and the assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and<answer> </answer> tags, respectively, i.e., <think> reasoning process here </think><answer> answer here </answer>.\n<|im_end|>\n<|im_start|>user\nLet $\\bar{X}$ be the mean of a random sample of size 12 from the uniform distribution on the interval $(0,1)$. Approximate $P(1 / 2 \\leq \\bar{X} \\leq 2 / 3)$.\n<|im_end|>\n<|im_start|>assistant\n<think>",
"role": "user"
}
]
| logic | {
"ground_truth": {
"answer_number": "0.4772",
"unit": " "
},
"style": "rule"
} | {
"index": 391,
"split": "train"
} |
|||
Determine the constant $c$ such that $f(x)= c x^3(1-x)^6$, $0 < x < 1$ is a pdf. | 840 | 840 | stat | 5.2-9 | scibench | [
{
"content": "<|im_start|>system\nA conversation between User and Assistant. The user asks a question, and the assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and<answer> </answer> tags, respectively, i.e., <think> reasoning process here </think><answer> answer here </answer>.\n<|im_end|>\n<|im_start|>user\nDetermine the constant $c$ such that $f(x)= c x^3(1-x)^6$, $0 < x < 1$ is a pdf.\n<|im_end|>\n<|im_start|>assistant\n<think>",
"role": "user"
}
]
| logic | {
"ground_truth": {
"answer_number": "840",
"unit": " "
},
"style": "rule"
} | {
"index": 392,
"split": "train"
} |
|||
Three drugs are being tested for use as the treatment of a certain disease. Let $p_1, p_2$, and $p_3$ represent the probabilities of success for the respective drugs. As three patients come in, each is given one of the drugs in a random order. After $n=10$ 'triples' and assuming independence, compute the probability that the maximum number of successes with one of the drugs exceeds eight if, in fact, $p_1=p_2=p_3=0.7$ | $0.0384$ | 0.0384 | stat | 5.3-15 | scibench | [
{
"content": "<|im_start|>system\nA conversation between User and Assistant. The user asks a question, and the assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and<answer> </answer> tags, respectively, i.e., <think> reasoning process here </think><answer> answer here </answer>.\n<|im_end|>\n<|im_start|>user\nThree drugs are being tested for use as the treatment of a certain disease. Let $p_1, p_2$, and $p_3$ represent the probabilities of success for the respective drugs. As three patients come in, each is given one of the drugs in a random order. After $n=10$ 'triples' and assuming independence, compute the probability that the maximum number of successes with one of the drugs exceeds eight if, in fact, $p_1=p_2=p_3=0.7$ \n<|im_end|>\n<|im_start|>assistant\n<think>",
"role": "user"
}
]
| logic | {
"ground_truth": {
"answer_number": "0.0384",
"unit": " "
},
"style": "rule"
} | {
"index": 393,
"split": "train"
} |
|||
Evaluate
$$
\int_0^{0.4} \frac{\Gamma(7)}{\Gamma(4) \Gamma(3)} y^3(1-y)^2 d y
$$ Using integration. | 0.1792 | 0.1792 | stat | 5.2-11 | scibench | [
{
"content": "<|im_start|>system\nA conversation between User and Assistant. The user asks a question, and the assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and<answer> </answer> tags, respectively, i.e., <think> reasoning process here </think><answer> answer here </answer>.\n<|im_end|>\n<|im_start|>user\nEvaluate\r\n$$\r\n\\int_0^{0.4} \\frac{\\Gamma(7)}{\\Gamma(4) \\Gamma(3)} y^3(1-y)^2 d y\r\n$$ Using integration.\n<|im_end|>\n<|im_start|>assistant\n<think>",
"role": "user"
}
]
| logic | {
"ground_truth": {
"answer_number": "0.1792",
"unit": " "
},
"style": "rule"
} | {
"index": 394,
"split": "train"
} |
|||
Let $X$ equal the maximal oxygen intake of a human on a treadmill, where the measurements are in milliliters of oxygen per minute per kilogram of weight. Assume that, for a particular population, the mean of $X$ is $\mu=$ 54.030 and the standard deviation is $\sigma=5.8$. Let $\bar{X}$ be the sample mean of a random sample of size $n=47$. Find $P(52.761 \leq \bar{X} \leq 54.453)$, approximately. | $0.6247$ | 0.6247 | stat | 5.6-7 | scibench | [
{
"content": "<|im_start|>system\nA conversation between User and Assistant. The user asks a question, and the assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and<answer> </answer> tags, respectively, i.e., <think> reasoning process here </think><answer> answer here </answer>.\n<|im_end|>\n<|im_start|>user\nLet $X$ equal the maximal oxygen intake of a human on a treadmill, where the measurements are in milliliters of oxygen per minute per kilogram of weight. Assume that, for a particular population, the mean of $X$ is $\\mu=$ 54.030 and the standard deviation is $\\sigma=5.8$. Let $\\bar{X}$ be the sample mean of a random sample of size $n=47$. Find $P(52.761 \\leq \\bar{X} \\leq 54.453)$, approximately.\n<|im_end|>\n<|im_start|>assistant\n<think>",
"role": "user"
}
]
| logic | {
"ground_truth": {
"answer_number": "0.6247",
"unit": " "
},
"style": "rule"
} | {
"index": 395,
"split": "train"
} |
|||
Two components operate in parallel in a device, so the device fails when and only when both components fail. The lifetimes, $X_1$ and $X_2$, of the respective components are independent and identically distributed with an exponential distribution with $\theta=2$. The cost of operating the device is $Z=2 Y_1+Y_2$, where $Y_1=\min \left(X_1, X_2\right)$ and $Y_2=\max \left(X_1, X_2\right)$. Compute $E(Z)$. | $5$ | 5 | stat | 5.3-19 | scibench | [
{
"content": "<|im_start|>system\nA conversation between User and Assistant. The user asks a question, and the assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and<answer> </answer> tags, respectively, i.e., <think> reasoning process here </think><answer> answer here </answer>.\n<|im_end|>\n<|im_start|>user\nTwo components operate in parallel in a device, so the device fails when and only when both components fail. The lifetimes, $X_1$ and $X_2$, of the respective components are independent and identically distributed with an exponential distribution with $\\theta=2$. The cost of operating the device is $Z=2 Y_1+Y_2$, where $Y_1=\\min \\left(X_1, X_2\\right)$ and $Y_2=\\max \\left(X_1, X_2\\right)$. Compute $E(Z)$.\n<|im_end|>\n<|im_start|>assistant\n<think>",
"role": "user"
}
]
| logic | {
"ground_truth": {
"answer_number": "5",
"unit": " "
},
"style": "rule"
} | {
"index": 396,
"split": "train"
} |
|||
If $X$ is a random variable with mean 33 and variance 16, use Chebyshev's inequality to find An upper bound for $P(|X-33| \geq 14)$. | $0.082$ | 0.082 | stat | 5.8-1 | scibench | [
{
"content": "<|im_start|>system\nA conversation between User and Assistant. The user asks a question, and the assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and<answer> </answer> tags, respectively, i.e., <think> reasoning process here </think><answer> answer here </answer>.\n<|im_end|>\n<|im_start|>user\nIf $X$ is a random variable with mean 33 and variance 16, use Chebyshev's inequality to find An upper bound for $P(|X-33| \\geq 14)$.\n<|im_end|>\n<|im_start|>assistant\n<think>",
"role": "user"
}
]
| logic | {
"ground_truth": {
"answer_number": "0.082",
"unit": " "
},
"style": "rule"
} | {
"index": 397,
"split": "train"
} |
|||
Suppose that the length of life in hours (say, $X$ ) of a light bulb manufactured by company $A$ is $N(800,14400)$ and the length of life in hours (say, $Y$ ) of a light bulb manufactured by company $B$ is $N(850,2500)$. One bulb is randomly selected from each company and is burned until 'death.' Find the probability that the length of life of the bulb from company $A$ exceeds the length of life of the bulb from company $B$ by at least 15 hours. | $0.3085$ | 0.3085 | stat | 5.5-9 (a) | scibench | [
{
"content": "<|im_start|>system\nA conversation between User and Assistant. The user asks a question, and the assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and<answer> </answer> tags, respectively, i.e., <think> reasoning process here </think><answer> answer here </answer>.\n<|im_end|>\n<|im_start|>user\nSuppose that the length of life in hours (say, $X$ ) of a light bulb manufactured by company $A$ is $N(800,14400)$ and the length of life in hours (say, $Y$ ) of a light bulb manufactured by company $B$ is $N(850,2500)$. One bulb is randomly selected from each company and is burned until 'death.' Find the probability that the length of life of the bulb from company $A$ exceeds the length of life of the bulb from company $B$ by at least 15 hours.\n<|im_end|>\n<|im_start|>assistant\n<think>",
"role": "user"
}
]
| logic | {
"ground_truth": {
"answer_number": "0.3085",
"unit": " "
},
"style": "rule"
} | {
"index": 398,
"split": "train"
} |
|||
An urn contains 10 red and 10 white balls. The balls are drawn from the urn at random, one at a time. Find the probability that the fourth white ball is the fourth ball drawn if the sampling is done with replacement. | $\frac{1}{16}$ | 0.0625 | stat | Problem 1.4.15 | scibench | [
{
"content": "<|im_start|>system\nA conversation between User and Assistant. The user asks a question, and the assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and<answer> </answer> tags, respectively, i.e., <think> reasoning process here </think><answer> answer here </answer>.\n<|im_end|>\n<|im_start|>user\nAn urn contains 10 red and 10 white balls. The balls are drawn from the urn at random, one at a time. Find the probability that the fourth white ball is the fourth ball drawn if the sampling is done with replacement.\n<|im_end|>\n<|im_start|>assistant\n<think>",
"role": "user"
}
]
| logic | {
"ground_truth": {
"answer_number": "0.0625",
"unit": " "
},
"style": "rule"
} | {
"index": 399,
"split": "train"
} |
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