Optimization problem?

Question

A company must manufacture a closed rectangular box with a square base. The volume must be 980 cubic inches. The top and the bottom squares are made of a material that costs 8 dollars per square inch. The vertical sides are made of a different material that costs 5 dollars per sqare inch.
What is the minimal cost of a box of this type?

Answer 1

I call x the edge of the square

The area of the top and the bottom is x² + x² = 2x²

so the cost for the top and the bottom is 16x²

The height of the box is h and we have x²h = 980,
so h = 980/x²
and the area of the vertical sides is 4*x*h = 3920/x

The cost for the vertical sides is
5*3920/x = 19600/x

The total cost is
f(x) = 16x² + 19600/x

f '(x) = 32x - 19600/x² = (32x³ - 19600) / x²

the minimal cost is for f '(x) = 0, so for
x = (19600/32)^(1/3) = 612.5^(1/3)

(x = 8.49)