Calculus word problem

Question

You are the manager of a delivery company and you want wooden crates with an open top and a square base. The volume of each crate needs to be exactly 62500cm³ and as the manager you want the cost of material to be minimized as well. Find the demensions of the crate to minimize the amount of material.

Answer 1

Let V=62500

V=hs² where h is height and s is side of square

h = V/s²

The amount of material needed is 4hs + s² = 4V/s + s²

The derivative of the amount of material is -4V/s² +2s and is 0 when s = (2V)^1/3

^{You can do the actual calculation!}