How much longer does it take to paint a similar box with $5$ times more volume?

Question

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.

Answer 1

Suppose that the linear measurements of the large box are $k$ times the linear measurements of the small box. This would mean $3$-dimensional measurements, like volume, of the large box will be $k^3$ times those of the small box. So $100k^3 = 500$, and we see that $k = \sqrt[3]{5}$. The amount of surface area of a box to be spray painted is a $2$-dimensional measurement, so it will scale by $k^2 = \sqrt[3]{25}$. Since there is this much more surface area to spray paint, it will take this much more time to paint it, and so it'll take two people $10\sqrt[3]{5}$ minutes to paint the larger box. Dividing the work among three people instead of two, we see that to paint the large box it'll take $$ \frac{2}{3}10\sqrt[3]{25} \text{ minutes.} $$

Answer 2

For similar rectangular boxes, area ratio is the square of length ratio and volume ratio the cube of length ratio. So $5$ times the volume is $5^{\frac{2}{3}}$ times the area which is equivalent to $5^{\frac{2}{3}}$ boxes.

If it takes two people $10$ minutes to spray paint a box then the rate is $20$ minutes per person per box

For three people, it will take $\frac{20}{3} \cdot 5^{\frac{2}{3}}$ minutes to paint the $500$ cubic foot box.

$T = 19.4934$ minutes or $19$ min $29.6$ sec.

Answer 3

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