# optimazation problem - cylinder

• Apr 21st 2010, 12:24 PM
lsanger
optimazation problem - cylinder
hi,i wqas having a hard time solving this problem...any help would be great

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.
• Apr 21st 2010, 12:59 PM
Soroban
Hello, lsanger!

Quote:

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 $L\times W$ rectangle.

Code:

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

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

It is "rolled" into a cylinder.
The side view looks like this:

Code:

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

The circumference of the circular base is $L.$
. . $2\pi r \:=\:L \quad\Rightarrow\quad r \:=\:\frac{L}{2\pi}$ .[2]

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

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

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

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

And that is the function we must maximize.