Chapter 2.6, Problem 12E

An Aluminum Can The cost of making a can is determined by how much aluminum A, in square inches, is needed to make it. This in turn depends on the radius r and the height h of the can, both measured in inches. You will need some basic facts about cans. See Figure 2.107.

The surface of a can may be modeled as consisting of three parts: two circles of radius r and the surface of a cylinder of radius r and height h. The area of each circle is $\pi r^2$.

In what follows, we assume that the can must hold 15 cubic inches, and we will look at various cans holding the same volume.

a. Explain why the height of any can that holds a volume of 15 cubic inches is given by $h=\frac{15}{\pi r^2}$

b. Make a graph of the height h as a function of r, and explain what the graph is showing.

c. Is there a value of r that gives the least height h? Explain.

d. If A is the amount of aluminum needed to make the can, explain why $A=2\pi r^2+2\pi rh$.

e. Using the formula for h from part a, explain why we may also write A as $A=2\pi r^2+\frac{30}{r}$.