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 rectangle.

Code:

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W | |
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L

The perimeter is 12: . .[1]

It is "rolled" into a cylinder.

The side view looks like this:

Code:

* - + - *
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h | : |
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r

The circumference of the circular base is

. . .[2]

The height of the cylinder is .[3]

The volume of a cylinder is: .

Substitute [2] and [3]: .

Substitute [1]: .

And *that* is the function we must maximize.