Hello, lsanger!

Consider a rectangle of perimeter of 12 inches.

We obtain a cylinder by gluing vertical side to the other vertical side.

Find the dimensions of the rectangle so that the cylinder has maximal volume.

We have an $\displaystyle L\times W$ rectangle.

Code:

* - - - - - - - - *
| |
| |
W | |
| |
| |
* - - - - - - - - *
L

The perimeter is 12: .$\displaystyle 2L + 2W \:=\:12 \quad\Rightarrow\quad W \:=\:6-L$ .[1]

It is "rolled" into a cylinder.

The side view looks like this:

Code:

* - + - *
| : |
| : |
h | : |
| : |
| : |
* - + - *
r

The circumference of the circular base is $\displaystyle L.$

. . $\displaystyle 2\pi r \:=\:L \quad\Rightarrow\quad r \:=\:\frac{L}{2\pi}$ .[2]

The height $\displaystyle h$ of the cylinder is $\displaystyle W.$ .[3]

The volume of a cylinder is: .$\displaystyle V \;=\;\pi r^2h$

Substitute [2] and [3]: .$\displaystyle V \;=\;\pi\left(\frac{L}{2\pi}\right)^2W \;=\;\frac{1}{4\pi}L^2W$

Substitute [1]: .$\displaystyle V \;=\;\frac{1}{4\pi}L^2(6-L) \quad\Rightarrow\quad V \;=\;\frac{1}{4\pi}(6L^2-L^3) $

And *that* is the function we must maximize.