Search the web $\newline$ $2$ . $3$ Midterm $\newline$ //courses/ $19992$ /quizzes/ $103757$ /take $\newline$ Question $1$ $\newline$ $3$ pts $\newline$ The heat experienced by a hiker at a campfire is proportional to the amount of wood on the fire, and inversely proportional to the cube of his distance from the fire. If he is $19 \mathrm{ft}$ from the fire, and someone doubles the amount of wood burning, approximately how far from the fire would he have to be so that he feels the same heat as before? $\newline$ $38 \mathrm{ft}$ $\newline$ $13 \mathrm{ft}$ $\newline$ $28.5 \mathrm{ft}$ $\newline$ $90 \mathrm{ft}$ $\newline$ $24 \mathrm{ft}$

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Grade 7

Multiply and divide positive and negative fractions

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Q. Search the web

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2

3

Midterm

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//courses/

19992

/quizzes/

103757

/take

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Question

1

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3

pts

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The heat experienced by a hiker at a campfire is proportional to the amount of wood on the fire, and inversely proportional to the cube of his distance from the fire. If he is

19 \mathrm{ft}

from the fire, and someone doubles the amount of wood burning, approximately how far from the fire would he have to be so that he feels the same heat as before?

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38 \mathrm{ft}

\newline

13 \mathrm{ft}

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28.5 \mathrm{ft}

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90 \mathrm{ft}

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24 \mathrm{ft}

Define Variables: Let's call the initial amount of wood $W$ and the initial distance $D$ . The heat felt is proportional to $\frac{W}{D^3}$ . If the wood amount is doubled, the new amount of wood is $2W$ .
Proportional Heat Equation: To feel the same heat, the new distance ＂ $D_{\text{new}}$ ＂ must satisfy the equation $\frac{W}{D^3} = \frac{2W}{D_{\text{new}}^3}$ .
Simplify Equation: Simplify the equation to find $D_{\text{new}}$ : $\frac{D^3}{D_{\text{new}}^3} = 2$ .
Cube Root Calculation: Take the cube root of both sides to solve for $D_{\text{new}}$ : $D_{\text{new}} = D \times \sqrt[3]{2}$ .
Calculate New Distance: Calculate $D_{\text{new}}$ using the initial distance $D = 19\,\text{ft}$ : $D_{\text{new}} = 19 \times \sqrt[3]{2}$ .
Approximate and Multiply: Approximate the cube root of $2$ as $1.26$ (since $1.26^3 \approx 2$ ) and multiply by $19$ : $D_{\text{new}} \approx 19 \times 1.26$ .