Rational Function Question - Volume

Question

a right circular cylinder is to have a volume of 50 cubic inches. It costs 4 cents per square inch to do trust the top and bottom, and 1 cent per square inch to construct the rest of the cylinder. Find the radius to yeild minimum cost. Let x= radius.

Answer 1

For a right circular cylinder V = πr²h therefore 50 = πr²h or, in our case where x = the radius then 50 = πx²h which means h = 50/(πx²)

When they talk about "4 cents per square inch to do trust the top and bottom, and 1 cent per square inch to construct the rest of the cylinder" they are talking about the surface area (SA).

For a right cylinder, the surface area is calculated form:

SA = 2πrh + 2πr² or, in our case: SA = 2πxh + 2πx²

Plugging in "50/(πx²)" in place of "h" we get:

SA = 2πx(50/(πx²)) + 2πx² = 100/x + 2πx² therefore:

SA(x) = 100/x + 2πx² where the first term is the surface area of the sides and the second term is the surface area of the top and bottom.

Since the cost is 4 cents for the top and bottom and 1 cent for the sides, then:

C(x) = (0.01)100/x + (0.04)2πx²

C(x) = 1/x + 0.08πx²

The minimum can be found by graphing this function and looking for the local minimum. (Using calculus, you can take the derivative and set it equal to zero but since this question is listed under pre-calc, I will assume you need to look at the graph)

The graph looks like this:

The minimum cost occurs when x (the radius) is 1.258 inches