Question

A rectangle of perimeter 18 in. is rotated about one of its sides to generate a right circular cylinder. Find the rectangle which generates the cylinder of largest volume has area in square inches.

Answer 1

The volume of the cylinder from the rectangle is the size of the cylinder

The rectangle that generates the cylinder of the largest volume is 3 inches by 6 inches

How to determine the rectangle

Let the dimension of the rectangle be x and y.

So, the perimeter (P) is:

$P = 2 * (x + y)$

The perimeter is given as 18.

So, we have:

$2 * (x + y) = 18$

Divide both sides by 2

$x + y = 9$

Make y the subject

$y = 9 - x$

The volume of a cylinder is:

$V = \pi r^2h$

So, we have:

$V = \pi x^2(9 -x)$

Expand

$V = \pi(9x^2 -x^3)$

Differentiate

$V' = \pi(18x -3x^2)$

Set to 0

$\pi(18x -3x^2) = 0$

Divide both sides by $\pi$

$18x -3x^2 = 0$

Divide both sides by 3x

$6 -x = 0$

Make x the subject

$x = 6$

Recall that:

$y = 9 - x$

So, we have:

$y = 9 - 6$

$y = 3$

Hence, the rectangle that generates the cylinder of the largest volume is 3 inches by 6 inches

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