A box with a rectangular base and no top is to be made from a 9 inch by 12 inch piece of cardboard by cutting squares out of the corners and folding up the sides. What size (side-length) squares should be cut out to make the box have as much volume as possible?

Answer 1

If the squares have side x, then the volume is

v = x(9-2x)(12-2x) = 4x^3 - 42x^2 + 108x
dv/dx = 12x^2 - 84x + 108 = 12(x^2-7x+9)

now just find where dv/dx=0