A box with a square base and open top must have a volume of 4000 cm3. Find the dimensions of the box that minimize the amount of material used.

Question

A box with a square base and open top must have a volume of 4000 cm3. Find the dimensions of the box that minimize the amount of material used.

Answer 1

Let length = x

Let height = h

 

Volume of box is

 

x²h = 4000                       eq1

 

 

Surface area of box is

 

SA = x² + 4xh                       eq2

 

 

Substitute eq1 into eq2.

 

SA = x² + 4x(4000/x²)

 

SA = x² + 16000x^-1

 

 

Set the derivative of SA equal to zero and solve for x.