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?

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