Question

A box is contructed out of two different types of metal. the metal for the top and bottom, which are both square, costs $5 per square foot and the metal for the sides costs $4 per square foot. find the dimensions that minimize cost if the box has a volume of 15 cubic feet.

Answer 1

Total cost, C = 8x^2 + 12xz 
volume, V = (x^2)*z = 10 
C = 8x^2 + 12x*10/x^2 
= 8x^2 + 120/x 
dC/dx = 16x - 120/x^2 = 0 
16x = 120/x^2 
x^3 = 120/16
 x = 1.957 ft

d^2C/dx^2 = 16 +240/x^3 = +ve for x = 1.957 
so, C is minimum when
x = 1.957 ft z = 2.61 ft

Length of the base x is 1.957 ft and height of side z is 2.61 ft