A storage shed is to be built in the shape of a box with a square base. It is to have a volume of 343 cubic feet. The concrete for the base costs $2 per square foot, the material for the roof costs $3 per square foot, and the material for the sides costs $2.50 per square foot. Find the dimensions of the most economical shed.

x = 7 ft

h = 7 ft

Step-by-step explanation:

Let call "x " side of the square base and " h " the height of the shed, then

V (s) = x² * h ⇒ 343 = x² * h ⇒ h = 343 / x²

Total cost of the shed is

C (t) = Cost of the base (C₁) + Cost of the roof (C₂) + Cost of 4 sides (C₃)

Then

C₁ = 2*x² C₂ = 3*x² and C₃ = 4*2,50*x*h

In C₃ as h = 343 / x² Then C₃ = 10 * 343/x C₃ = 3430 / x

Total cost of the shed as a function of x is:

C (x) = 2*x² + 3*x² + 3430 / x (1)

Taking derivatives on both sides of the equation we get:

C' (x) = 10*x - 3430 / x²

C' (x) = 0 ⇒ 10*x - 3430 / x² = 0

x - 343 / x² = 0 ⇒ x³ = 343

x = ∛ 343

x = 7 ft

If we plug this value in equation 1 we see C (x) > 0 then for x = 7, we get a minimun in the function

h = 343 / x²

h = 7 ft