The surface area of a closed cylinder is 100 cm^2. What is the height of the cylinder that can be created using the same material and which has maximum volume?

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The surface area of a closed cylinder is 100 cm^2. The material used to build this cylinder is used to create another cylinder such that the volume is maximized.

Let the radius of the ends of the new cylinder be r and the height is h.

`2*pi*r^2 + 2*pi*r*h = 100`

=> `h = (100 - 2*pi*r^2)/(2*pi*r)`

The volume of this cylinder is `V = pi*r^2*h = (pi*r^2*(100 - 2*pi*r^2))/(2*pi*r)`

= `r*(50 - pi*r^2)`

= `50*r - pi*r^3`

To maximize V, determine the value of r for which `(dV)/(dr) = 0`

`50 - pi*3*r^2 = 0`

=> `r^2 = 50/(3*pi)`

=> `r = sqrt(50/(3*pi))`

This gives `h = (100 - 2*pi*(sqrt(50/(3*pi)))^2)/(2*pi*(sqrt(50/(3*pi))))`

= `(100 - 2*pi*(50/(3*pi)))/(2*pi*(sqrt(50/(3*pi))))`

= `(50 - 50/3)/sqrt((50*pi)/3)`

`~~ 4.606`

**The height of the required cylinder is approximately 4.606 cm**

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