Minimizing cost of a open box

Question

A rectangular box, made with a square base and no top, is to have a volume of 12 cubic feet. The material for the base of the box costs 9 dollars per square foot, and the material for the sides costs 4 dollars per square foot. Determine the minimum cost for this box.

Answer 1

Alright, let's set up some formulas. The volume of the box is L²xH = 12. Then the cost of the box is 9 times the bottom (9xL²) and 4 times the four sides (4x4xLxH). I'm going to try and eliminate height from this cost function by stating that H = 12/(L²). So

C(L) = 9L² + 16L(12/(L²))

C(L) = 9L² + 192/L

Now to optimize this, we take the derivative and set it equal to zero.

C'(L) = 18L - 192/L² = 0

18L = 192/L²

L³ = 32/3

L = √(32/3) ≈ 2.2013

Now plug that back in to find H.

H = 12/L² ≈ 2.4764

So the box is approximately 2.2 feet square for the base and is about 2.48 feet deep.