Question

We have a half cylinder roof of radius `r` and height `r` mounted on top of four rectangular walls of height `h`. We have `200π` `m^2` of plastic sheet to be used in the construction of this structure. What is the value of `r` that allows maximum volume?

Answer 1

`r=20/sqrt(3)=(20sqrt(3))/3`

Explanation:

Let me restate the question as I understand it.
Provided the surface area of this object is `200pi`, maximize the volume.

Plan
Knowing the surface area, we can represent a height `h` as a function of radius `r`, then we can represent volume as a function of only one parameter - radius `r`.
This function needs to be maximized using `r` as a parameter. That gives the value of `r`.

Surface area contains:
4 walls that form a side surface of a parallelepiped with a perimeter of a base `6r` and height `h`, which have total area of `6rh`.
1 roof, half of a side surface of a cylinder of a radius `r` and hight `r`, that has area of `pi r^2`
2 sides of the roof, semicircles of a radius `r`, total area of which is `pi r^2`.

The resulting total surface area of an object is
`S = 6rh+2pi r^2`
Knowing this to be equal to `200pi`, we can express `h` in terms of `r`:
`6rh+2pir^2=200pi`
`r=(100pi-pir^2)/(3r) = (100pi)/(3r) - pi/3r`#

The volume of this object has two parts: Below the roof and within the roof.

Below the roof we have a parallelepiped with area of the base `2r^2` and height `h`, that is its volume is
`V_1 = 2r^2h=200/3pir - 2/3pir^3`

Within the roof we have half a cylinder with radius `r` and height `r`, its volume is
`V_2 = 1/2pir^3`

We have to maximize the function
`V(r) = V_1+V_2 = 200/3pir - 2/3pir^3 + 1/2pir^3 = 200/3pir - 1/6pir^3`
that looks like this (not to scale)
graph{2x-0.6x^3 [-5.12, 5.114, -2.56, 2.56]}

This function reaches its maximum when it's derivative equals to zero for a positive argument.

`V'(r) = 200/3pi - 1/2pi r^2`

In the area of `r>0` it's equal to zero when `r=20/sqrt(3)=20sqrt(3)/3`.
That is the radius that gives the largest volume, given the surface area and a shape of an object.