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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
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
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