Q:

Compose a function based on a word problem

I'm not looking for answers, I'm just having a hard time with composing a function out of a Max/Min problem like this. Possibly just show me how you would compose the function and leave the rest for me?

Create a cylindrical can that minimizes the cost of materials but must hold 100 cubic
  inches. The top and bottom of the can cost 14 cents per square inch, while the sides cost only 7 cents per square inch.

A:

Let $V$ be the volume of the can and $A$ be the surface area of the can, with radius $r$ and height $h$

The Area of the can is described in the problem as being made from an identical top and bottom with a cost of 14 and a side of cost 7.
$$V=100$$
$$A=2(top) + (side)$$
$$A=2(\pi r^2) + 2\pi rh$$
As a cost function:
$$C=14(2\pi r^2) + 7(2\pi rh)\tag{1}$$
and the known volume can be equated to the area by a factor of the height:
$$V=\pi r^2h=100$$
$$\Rightarrow h=\frac{100}{\pi r^2}$$
Substitute into $(1)$
$$C=14(2\pi r^2) + 7(2\pi r\frac{100}{\pi r^2})$$
Your units remain as cents per square inch. Simplify and differentiate.

