A box with a square base and open top must have a volume of 32000 cm3. Find the dimensions of the box that maximize the amount of material used.

Answer 1

First of all, it should be minimize the amount of materials, please check it.

Let x be the base length/width and h be the height.

V=x^2*h=32000==> h=32000/x^2

The function you want to minimize is the surface area

S=x^2+4xh=x^2+4x*(32000/x^2)=x^2+128000/x

Take the derivative and set it equal to 0

dS/dx=2x-128000/x^2=0

2x=128000/x^2

x^3=64000

x=40

h=32000/x^2=20

The first or second derivative test will show that these dimensions will minimize the amount of material used.