Another minimizing cost calc problem..help please..?

Question

A company must manufacture a closed rectangular box with a square base. The volume must be 1040 cubic inches. The top and the bottom squares are made of a material that costs 7 dollars per square inch. The vertical sides are made of a different material that costs 6 dollars per sqare inch.
What is the minimal cost of a box of this type?

Answer 1

Let s denote the length (in inches) of a side of the square base.
Let h denote the height (in inches) of the height of the box.

The volume V equals length × width × height so

V = s² h

and this must be 1040, so

s² h = 1040 . Solving for h in terms of s,

h = 1040/s²

The cost of the top and bottom sides (which are squares of side s, and hence of area s²) equals the area times the cost per square inch. So the cost of the material for the top is

7s²

and similarly for the bottom.

The vertical sides are rectangles of size s by h, of area s×h; so the cost of the material for each vertical side is

6sh

The total cost C of the material is given by

C = 2*7s² + 4*6sh = 14s² + 24sh

Using the expression obtained above for h in terms of s,

C = 14s² + 24s(1040/s²) = 14s² + 24960/s

I leave it to you to minimize C.