**
Problem**

You want to smuggle a precious metal out of the country, by disguising it as a single cylindrical barrel, closed at both ends. The cost of shipping is $7 per cubic foot. Once out the the country, you can sell the metal for $8 per square foot. Assuming that you design the barrels with the height equal to twice the diameter, how many square feet should you smuggle, and what will your profit be?

**
Solution**

For a cylinder of radius r and height h,

Since the height h is to be twice the diameter, and the diameter is twice the radius r,

and therefore we can write formulas for both the volume and the area in terms of a single variable, r.

The profit will be

As a function of the radius, the profit is

To maximize the profit, take its derivative, set it equal to zero, and solve for r.

This has solutions

The value r = 0 gives a barrel of zero volume and surface area, and a corresponding profit of zero, so it is unlikely to be the correct answer. With r = 40/21, we get

The number of square feet of metal needed to produce this profit is