# Thread: a maximum problem with cuboids!

1. ## a maximum problem with cuboids!

Hi, I'm new to this forum. I hope you can help me with this problem.

A block of wood in the shape of a cuboid is to have square ends. What is the volume of the largest block if the sum of the length of the block and teh perimeter of the end is 12cm?

Thanks!

2. Welcome to the forum!

We begin by stating the information given in the problem in mathematical terms. We'll call the perimeter $P$ and the length $L$. We know, first of all, that

$P+L=12.$

We may use this to substitute $P$ for $L$ at any time:

$L=12-P.$

We also know that since $\frac{P}{4}$ is the length of one side of the square end, the volume of the cuboid will be

$V=\mbox{area of square end}\cdot\mbox{length}=\left(\frac{P}{4}\right)^2\ cdot (12-P).$

To find the maximum volume, we note that $V=0$ at the endpoints of the domain (where $P$ or $L = 0$) and that $V$ is differentiable everywhere. Therefore, $V$ will attain a maximum at some point where $\frac{dV}{dP}=0$.