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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.

Sagot :

MrRoyal

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:

[tex]P = 2 * (x + y)[/tex]

The perimeter is given as 18.

So, we have:

[tex]2 * (x + y) = 18[/tex]

Divide both sides by 2

[tex]x + y = 9[/tex]

Make y the subject

[tex]y = 9 - x[/tex]

The volume of a cylinder is:

[tex]V = \pi r^2h[/tex]

So, we have:

[tex]V = \pi x^2(9 -x)[/tex]

Expand

[tex]V = \pi(9x^2 -x^3)[/tex]

Differentiate

[tex]V' = \pi(18x -3x^2)[/tex]

Set to 0

[tex]\pi(18x -3x^2) = 0[/tex]

Divide both sides by [tex]\pi[/tex]

[tex]18x -3x^2 = 0[/tex]

Divide both sides by 3x

[tex]6 -x = 0[/tex]

Make x the subject

[tex]x = 6[/tex]

Recall that:

[tex]y = 9 - x[/tex]

So, we have:

[tex]y = 9 - 6[/tex]

[tex]y = 3[/tex]

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

Read more about volumes at:

https://brainly.com/question/10171109

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