Question

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.

Answer 1

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)
 Answer:
 The dimensions for minimum cost are:
 40 * 40 * 8.75