You are under contract to design a storage building with a square base and a volume of 14,000 cubic feet. the cost of materials is $4 per square foot for the floor, $16 per square foot for the walls and $3 per square foot for the roof. find the dimensions that minimize the cost of materials.

The first thing we are going to do for this case is define variables.
 We have then:
 y = the cost of the box
 x = one side of the square base
 z = height of the box
 The volume of the building is 14,000 cubic feet:
 x ^ 2 * z = 14000
 We cleared z:
 z = (14000 / x ^ 2)
 On the other hand, the cost will be:
 floor = 4 (x ^ 2)
 roof = 3 (x ^ 2)
 for the walls:
 1 side = 16 (x * (14000 / x ^ 2)) = 16 (14000 / x)
 4 sides = 64 (14000 / x) = 896000 / x
 The total cost is:
 y = floor + roof + walls
 y = 4 (x ^ 2) + 3 (x ^ 2) + 896000 / x
 y = 7 (x ^ 2) + 896000 / x
 We derive the function:
 y '= 14x - 896000 / x ^ 2
 We match zero:
 0 = 14x - 896000 / x ^ 2
 We clear x:
 14x = 896000 / x ^ 2
 x ^ 3 = 896000/14
 x = (896000/14) ^ (1/3)
 x = 40
 min cost (y) occurs when x = 40 ft
 Then,
 y = 7 * (40 ^ 2) + 896000/40
 y = 33600 $
 Then the height
 z = 14000/40 ^ 2 = 8.75 ft
 The price is:
 floor = 4 * (40 ^ 2) = 6400
 roof = 3 * (40 ^ 2) = 4800
 walls = 16 * 4 * (40 * 8.75) = 22400
 Total cost = $ 33600 (as calculated previously)
 
 The dimensions for minimum cost are:
 40 * 40 * 8.75