Calculus I Optimization Problem

Question

A square-based, box-shaped shipping crate with a volume of 16 cubic feet needs to be constructed at minimum cost. The material used for the sides of the box costs p dollars per square foot, while the material used for each square base (top and bottom of the crate) costs 2p dollars per square foot. What are the dimensions of the crate that minimize the cost of materials? Show all steps. Use appropriate units. Remember to use either the First Derivative Test or the Second Derivative Test to show that the cost has been minimized, and label the dimensions of the optimized shipping crate.

Hi! I am having trouble with this problem. I know how to solve these when the cost is known, but this problem gives cost as p and 2p, and I'm not sure how to work with that properly. Thanks!

Answer 1

h=4p height = $4 times p

b=2p base side = $2 times p

V=16 = hb^2

h= 16/b^2

Cost = C = 4b^2 + 4b(16/b^2)

C= 4b^2 + 64/b

C' = 8b -64/b^2 = 0

8b^3 - 64 = 0

b^3 = 8

b = 2

h= 16/4 = 4

height is twice the base side if cost is twice as much for the base side