Find the dimensions that minimize cost of the box

Question

A box is constructed 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 $2 per square foot. Find the dimensions that minimize cost if the box has a volume of 50 cubic feet.

Length of base x=

Height of side z=

Answer 1

box volume = 50 cubic feet

cost of square top & bottom = $5 per ft^2

cost of 4 sides = $2 per ft^2

what are dimensions of least costly box?? = about 3.4 x 3.4 x 4.2 feet

50 = hs^2, h = 50/s^2

V = 50 = hs^2

cost = 2h + 5s = 100/s^2 + 5s

take the derivative of Cost and set = 0, solve for side length of top and bottom

C' = -200/s^3 + 5 = 0

40/s^3 = 1

s^3 = 40

s = about 3.42 feet

h = 50/s^2 = about 4.27 feet

dimensions of least costly box

are about 3.42 x 3.42 x 4.27 feet