A box with a square base and open top must have a volume of 32,000 cm^2. Find the dimensions of the box that minimize the amount of material used.

Question

optimization problems or kinda problem solving. thank you in advance

Answer 1

Let x = length of a side of the base

 

     y = height

 

Then, x²y = 32000, so y = 32000 / x²

 

Let S = surface area = area of base + total area of the 4 sides

 

                               = x² + 4xy = x² + 4x(32000 / x²)

 

                                                = x² + 128,000 / x, x > 0

 

Minimize S:  S' = 2x - 128,000 / x²

 

                      = (2x³ - 128,000) / x²

 

                  S' = 0 when 2x³ = 128,000

 

                                      x³ = 64,000

 

                                      x = 40

 

When 0 < x < 40, S' < 0.  So, S is decreasing.

 

When x > 40, S' > 0, so S is increasing.

 

Therefore, surface area is minimized when x = 40 cm and y = 32000 / 40² = 20 cm.