A storage shed is to be built in the shape of a box with a square base. It is to have a volume of 250 cubic feet. The concrete for the base costs $4 per square foot, the material for the roof costs $6 per square foot, and the material for the sides costs $2.50 per square foot. Find the dimensions of the most economical shed.

the dimensions of the most economical shed are height = 10 ft and side 5 ft

Step-by-step explanation:

Given data

volume = 250 cubic feet

base costs = $4 per square foot

material for the roof costs = $6 per square foot

material for the sides costs = $2.50 per square foot

to find out

the dimensions of the most economical shed

solution

let us consider length of side x and height is h

so we can say x²h = 250

and h = 250 / x²

now cost of material = cost of base + cost top + cost 4 side

cost = x² (4) + x² (6) + 4xh (2.5)

cost = 10 x² + 10xh

put here h = 250 / x²

cost = 10 x² + 10x (250 / x²)

cost = 10 x² + (2500 / x)

differentiate and we get

c' = 20 x - 2500 / x²

put c' = 0 solve x

20 x - 2500 / x² = 0

x = 5

so we say one side is 5 ft base

and height is h = 250 / x²

h = 250 / 5²

height = 10 ft