Question

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.

Answer 1

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.