How do I solve this calculus optimization question?

Question

The volume of a storage facility must be 800m3. The shape of this building must be a rectangular prism with length double the width. If the floor costs $50/m2, the walls cost $100/m2, and the roof (which is flat) costs $250/m2, what dimensions will create the lowest total cost?

Answer 1

Let w=width, then length=2w. It is assumed that the length and width are those of the floor and the roof. Let h=height (of the prism).

Volume 800=2w²h, h=400/w².

Area of floor and roof=2w², cost of roof=$500w², cost of floor=$100w².

Cost of floor and roof=$600w².

Area of walls=4wh+2wh=6wh,

cost of walls=$600wh=600w(400/w²)=$240000/w.

Total cost C=240000/w+600w².

dC/dw=-240000/w²+1200w=(-240000+1200w³)/w².

When dC/dw=0, 

1200w³=240000, w³=240000/1200=200.

Therefore, width w=∛200=5.8480m approx.

Length=11.6961m, height=400/w²=11.6961m.

Minimum cost=$61,559.13.