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 567567 cubic feet. The concrete for the base costs ​$22 per square​ foot, the material for the roof costs ​$55 per square​ foot, and the material for the sides costs ​$4.504.50 per square foot. Find the dimensions of the most economical shed.

Answer 1

length of the base = 9 ft

and height of the shed = 7 ft

Explanation:

given data

volume = 567 cubic feet

base costs = ​$2 per square​ foot

roof costs ​= $5 per square​ foot

sides costs ​$4.50 per square foot

solution

we take here length of the base = x ft

and height of the shed = y ft

so Volume will be express as

volume = x²× y

567 = x² × y

y =
`(567)/(x^2)`

and

we know cost of material is express as here

cost of material = cost of base + cost top + cost 4 side ..................1

put here value

cost = x²(2) + x²(5) + 4xy (4.5)

cost = 7x² + 18xy

put here y value

cost = 7x² + 18 x (
`(567)/(x^2)` )

differentiate and we get

C' = 14x -
`(10206)/(x^2)`

we put here C' = 0 and we get

14x -
`(10206)/(x^2)` = 0

solve it we get

x = 9 ft

and

y =
`(567)/(9^2)`

y = 7 ft