Question

A storage box with a square base must have a volume of 80 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.) x

Answer 1

Box dimensions:

x = 3.42 cm

y = 6.84 cm

C(min) = 14.04 $

Explanation:

We need the surface area of the cube:

S(c) = 2*S₁ ( surface area of top or base) + 4*S₂ ( surface lateral area)

S₁ = x² 2*S₁ = 2*x²

Surface lateral area is:

4*S₂ = 4*x*h V(c) = 80 cm³ = x²*h h = 80/x²

4*S₂ = 4*80/x

4*S₂ = 320 / x

Costs

C (x) = 0.2* 2*x² + 0.1 * 320/x

Taking derivatives on both sides of the equation we get:

C´(x) = 0.8*x - 32/x²

C´(x) = 0 0.8*x - 32/x² = 0

0.8*x³ - 32 = 0 x³ = 32/0.8

x³ = 40

x = 3.42 cm

h = 80/(3.42)² h = 6.84 cm

To find out if x = 3.42 brings a minimum value for C we go to the second derivative

C´´(x) = 64/x³ is always positive for x > 0

The C(min) = 0.4*(3.42)² + 32/(3.42)

C(min) = 4.68 + 9.36

C(min) = 14.04 $