College Algebra, Chapter 2, 2.5, Section 2.5, Problem 46

The heat $H$ experienced by a hiker at a campfire is proportional to the amount of wood $W$ on the fire and inversely proportional to the cube of his distance $s$ from the fire. If he is 20ft from the fire and someone doubles the amount of wood burning, how far from the fire would he have to be so that he feels the same heat before?

$\displaystyle H = \frac{kW}{s^3}$ model
Let $W_1$ and $s_1$ be the quantities involved in the initial scenario and $W_2$ and $s_2$ be the quantities on the next scenario. Since the problem stated that the hiker wants to feel the same heat as before we have,

$
\begin{equation}
\begin{aligned}
H &= \frac{kW_1}{(s_1)^3} &&\text{and}& H &= \frac{kW_2}{(s_2)^3} && \text{Recall that } W_2 = 2W_1\\
\\
\frac{H}{k} &= \frac{W_1}{(s_1)^3} &&\text{and}& \frac{H}{k} &= \frac{2W_1}{(s_2)^3} && \text{Equate both } \frac{H}{k}
\end{aligned}
\end{equation}
$



$
\begin{equation}
\begin{aligned}
\frac{W_1}{(s_1)^3} &= \frac{2W_1}{(s_2)^3} && \text{Cancel out like terms}\\
\\
\frac{1}{(s_1)^3} &= \frac{2}{(s_2)^3} && \text{Apply cross multiplication}\\
\\
(s_2)^3 &= 2(s_1)^3 && \text{Take the cube root}\\
\\
s_2 &= \sqrt[3]{2} s_1
\end{aligned}
\end{equation}
$

It shows that, in order for the hiker to feel the same heat, considering that the amount of woods double, the hiker must stay away from the fire at $s_2 = \sqrt[3]{2}$ $s_1 = \sqrt[3]{2} \left(20 \text{ft}\right) = 25.20 \text{ft}$.