TITLE: How much longer does it take to paint a similar box with $5$ times more volume?
QUESTION [6 upvotes]: This is a question from a precalculus class that I'm a TA for.

You build a box that has a volume of 100 cubic feet. It takes two people ten minutes to spray paint this box. How many minutes does it take three people to spray paint a similar box that has a volume of 500 cubic feet? Assume that the area of surface to be covered is proportionate to both the number of people working and the time spent spray painting.

A student just asked me how to answer it, so I figured I'd write up the correct calculations and post it online to help anyone else who may wander across it.

REPLY [3 votes]: Refer to the table:
$$\begin{array}{c|c|c|c}
\text{People} & \text{Time} & \text{Surface}\\
\hline
\color{blue}2 & \color{blue}{10} & \color{blue}{6\sqrt[3]{100^2}}&\text{divide by $2$}\\
\color{blue}1 & \color{blue}{10} & \color{blue}{3\sqrt[3]{100^2}}&\text{multuply by $3$}\\
\color{blue}3 & \color{blue}{10} & \color{blue}{9\sqrt[3]{100^2}}\\
\hline
\color{red}3 &  \color{red}x & \color{red}{6\sqrt[3]{500^2}}\\
\end{array} \Rightarrow \color{red}x=\frac{\color{red}{6\sqrt[3]{500^2}}\cdot \color{blue}{10}}{\color{blue}{9\sqrt[3]{100^2}}}=\frac{20\sqrt[3]{25}}{3}.$$