A closed box with a square base is to have a volume of 13 comma 500 cm cubed. The material for the top and bottom of the box costs $10.00 per square centimeter, while the material for the sides costs $2.50 per square centimeter. Find the dimensions of the box that will lead to the minimum total cost. What is the minimum total cost?

x = 1,5 cm

h = 6 cm

C (min) = 135 $

Step-by-step explanation:

Volume of the box is:

V (b) = 13,5 cm³

Aea of the top is equal to area of the base,

Let call " x " side of the base then as it is square area is A₁ = x²

Sides areas are 4 each one equal to x * h (where h is the high of the box)

And volume of the box is 13,5 cm³ = x²*h

Then h = 13,5/x²

Side area is : A₂ = x * 13,5/x²

A (b) = A₁ + A₂

Total area of the box as functon of x is:

A (x) = 2*x² + 4 * 13,5 / x

And finally cost of the box is

C (x) = 10*2*x² + 2.50*4*13.5/x

C (x) = 20*x² + 135/x

Taking derivatives on both sides of the equation:

C' (x) = 40*x - 135*/x²

C' (x) = 0 ⇒ 40*x - 135*/x² = 0 ⇒ 40*x³ = 135

x³ = 3.375

x = 1,5 cm

And h = 13,5/x² ⇒ h = 13,5 / (1,5) ²

h = 6 cm

C (min) = 20*x² + 135/x

C (min) = 45 + 90

C (min) = 135 $