Help with Applied Optimization

Question

a box is constructed out of two different types of metal. the metal for the top and bottom, which are both square, costs $4 per square foot and the metal for the sides costs $7 per square foot. find the dimensions that minimize cost if the box has a volume of 50 cubic feet. 

Answer 1

V=l^2h = 50 or h=50/l^2

Cost = 2l^2 * 4 + 4lh * 7

So that we don't have to use square roots initially, we substitute h =50/l^2 into the cost equation.

Minimize C=8l^2 + 28l(50/l^2) = 8l^2 + 1400/l

dC/dl = 16l - 1400/l²

16l - 1400/l² =0

16l³ = 1400

l = (175/2)^(1/3) = 4.4395

h = 50/l^2 = (800/49)^(1/3) = 2 *(100/49)^(1/3) = 2.5369

Note how l/h =7/4, the reciprocal of their cost.