# Problem of the Week Problem E and Solution Picking Boxes

## Problem

Billy’s Box Company sells boxes with the following very particular restrictions on their dimensions.

• The length, width, and height, in cm, must be all integers.

• The ratio of the length to the width to the height must be $$4:3:5$$.

• The sum of the length, width, and height must be between $$100$$ cm and $$1000$$ cm, inclusive.

Stefan bought the box with the smallest possible volume, and Lali bought the box with the largest volume less than $$4~\text{m}^3$$.

Determine the dimensions of Stefan and Lali’s boxes.

## Solution

Since the boxes from Billy’s Box Company have integer side lengths in the ratio $$4:3:5$$, let $$4n$$ represent the length of a box in cm, let $$3n$$ represent the width of a box in cm, and let $$5n$$ represent the height of a box in cm, where $$n$$ is an integer.

Furthermore, the sum of the length, width and height must be at least $$100$$ cm. It follows that \begin{aligned} 4n+3n+5n &\ge 100\\ 12n &\ge 100\\ n &\ge \frac{100}{12}=8 \frac{1}{3} \end{aligned}

Also, the sum of the length, width and height must be at most $$1000$$ cm. It follows that \begin{aligned} 4n+3n+5n &\le 1000\\ 12n &\le 1000\\ n &\le \frac{1000}{12}=83 \frac{1}{3} \end{aligned}

There is one other restriction to consider, since the volume of Lali’s box is less than $$4~\text{m}^3$$. To convert from $$\text{m}^3$$ to $$\text{cm}^3$$, note that \begin{aligned} 1~\text{m}^3 &= 1~\text{m} \times 1~\text{m} \times 1~\text{m}\\ &= 100~\text{cm} \times 100~\text{cm} \times 100~\text{cm}\\ &= 1\,000\,000~\text{cm}^3 \end{aligned} Therefore, $$4~\text{m}^3=4\,000\,000~\text{cm}^3$$.

It follows that \begin{aligned} (4n)(3n)(5n)&< 4\,000\,000\\ 60n^3 &< 4\,000\,000\\ n^3 &< \frac{200\,000}{3}\\ n &< \sqrt[3]{\frac{200\,000}{3}}\approx 40.5 \end{aligned}

We also know that $$n$$ is an integer. Since $$n\ge 8\frac{1}{3}$$, then the smallest possible integer value of $$n$$ is $$9$$. Using the dimensions $$4n,~ 3n,$$ and $$5n$$ with $$n=9$$, we can determine that the dimensions of Stefan’s box are $$36$$ cm by $$27$$ cm by $$45$$ cm.

For Lali’s box, since $$n\le 83\frac{1}{3}$$ and $$n<40.5$$, then the largest possible value of $$n$$ is $$40$$. Using the dimensions $$4n,~ 3n,$$ and $$5n$$ with $$n=40$$, we can determine that the dimensions of Lali’s box are $$160$$ cm by $$120$$ cm by $$200$$ cm. This box has a volume of $$3.84~\text{m}^3$$.