A storage shed is to be built in the shape of a box with a square base. it is to have a volume of 125125 cubic feet. the concrete for the base costs $88 per square foot, the material for the roof costs $99 per square foot, and the material for the sides costs $8.508.50 per square foot. find the dimensions of the most economical shed.

Let x represent the side length of the base.

The area of the base and roof are each x^2, so their cost together will be

... (8 + 9) x^2 = 17x^2

The height of the shed will be 125/x^2, so the area of one side will be

... (125/x^2) * x*8.50

and the area of all four sides will be

... 4*125/x*8.50 = 4250/x

So, the total cost is the cost of top and bottom together with the cost of the sides:

... cost = 17x^2 + 4250/x

This is minimized when its derivative with respect to x is zero.

... d (cost) / dx = 0 = 2*17x - 4250/x^2

... 4250/34 = x^3

... x = ∛125 = 5

The most economical shed is a 5 ft cube.