Question

A storage box with a square base must have a volume of 90 cubic centimeters. The top and bottom cost $0.20 per square centimeter and the sides cost $0.10 per square centimeter. Find the dimensions that will minimize cost. (Let x represent the length of the sides of the square base and let y represent the height. Round your answers to two decimal places.)

Answer 1

x = 2,24 cm

y = 17,96 cm

Explanation:

The volume of a cube is:

V = x²*y where x is the side of the square base and the height of the cube

V = 90 cm³

And the surface area of the cube is:

S(c) = Area of the base (A₁ ) + Lateral area (A₂)

A₁ = x² and V = x²*y then y = 90/x²

Therefore total cost is:

Cost of the top plus bottom C₁ = 0,20 * 2 * x² C₁ = 0,4*x²

Lateral cost (C₂) is C₂ = 0,10 * x*y then C₂ = 0,10*x ( 90/x²) C₂ = 9/x

C(x) = 0,4*x² + 9/x

We take derivatives on both sides of the equation to get:

C´(x) = 0,8*x - 9/x²

C´(x) = 0 0,8*x - 9/x² = 0

0,8*x³ - 9 = 0

0,8 * x³ = 9

x³ = 11,25

x = ∛ 11,25

x = 2,24 cm

And y = 90 / 5,01 y = 17,96 cm