Question

From a thin piece of cardboard 40 in by 40 in, square corners are cut out so that the sides can be folded up to make a box. what dimensions will yield a box of maximum volume? what is the maximum volume?

Answer 1

The dimensions of the square base after cutting the corners will be (40-2x). The depth of the box will be x, so its volume is given by
  V = x(40-2x)²

You can differentiate this to get
  V' = 12x² -320x -1600
Setting this to zero and factoring gives
  (3x-20)(x-20) = 0
The appropriate choice of solutions is
  x = 20/3 = 6 2/3

The dimensions of the box of maximum volume are
  26 2/3 in square by 6 2/3 in deep

The maximum volume is
  (80/3 in)²(20/3 in) = 4740 20/27 in³