Calculus - Optimization Problem?

Question

A customer wants to know the minimum cost of a water tank that will hold 750 gallons. Your company can make the tank from a rectangular piece of metal which can be rolled into a cylinder with two circular ends welded to the cylinder. The rectangular piece is made of a malleable alloy that costs $2.40 per square foot, but the circular ends can be made of a cheaper metal that costs $1.50 per square foot. Note, there are 7.5 gallons in a cubic foot. What dimensions ensure the lowest cost? What is the lowest cost?

Answer 1

This is pretty straightforward.
First you have to write the formula fo rthe cost of the tank as a function of, say, its radius, r:

The volume of the cylinder is 100 cubic feet.
It is also pi*r^2 times the height, h

so let's solve for h
100=pi*r^2*h
h = 100/(pi*r^2)

next, the cost:
the rectangle cost is 2.40*(h*2*pi*r),
or 2.4*100/(pi*r^2)*2*pi*r
or, simplifying, 480/r
The cost of the base and top are:
1.50*2 pieces * pi*r^2

So the total cost is
480/r+3*pi*r^2

Now you need to differentiate this and find the roots, and make sure they are minima, not maxima!