A rectangular box with a square base and top is to be made to contain 1250 cubic feet. The material for the base costs 35 cents per square foot, for the top 15 cents per square foot, and for the sides 20 cents per square foot. Find the dimensions that will minimize the cost of the box.
You would assume the side of the square (base or top) to be x (feet).
The height can be expressed in terms of x and the volume. (V=L*W*H=x²*H)
You would then express the sum of the costs, C, made up of
1. the cost of the base = area of base * unit cost of base,
2. the cost of the sides and
3. the cost of the top.
Now differentiate C(x) with respect to x and equate C'(x)=0. Solve for x.