# 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?

##### 1 Answer

#### 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

This function needs to be maximized using

Surface area contains:

4 walls that form a side surface of a parallelepiped with a perimeter of a base

1 roof, half of a side surface of a cylinder of a radius

2 sides of the roof, semicircles of a radius

The resulting total surface area of an object is

Knowing this to be equal to

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

Within the roof we have half a cylinder with radius

We have to maximize the function

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.

In the area of

That is the radius that gives the largest volume, given the surface area and a shape of an object.