IDNLearn.com: Your trusted source for accurate and reliable answers. Whether it's a simple query or a complex problem, our community has the answers you need.
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
Read more about volumes at:
https://brainly.com/question/10171109