I have a square piece of cardboard that is 24 inches on each side. i want to cut out a square in each corner and the fold up the sides to make an open box. how big should i make the squares that i cut out in order to make the box have the greatest volume?

A square of side 4 inch must be cut in order to get greatest volume.

after cutting a square of side x inch and folding it into a box, the length of each side of the box's base is (24-2x) inch and box's height is x inch.

∴volume, V = (24-2x) ² * x

differentiate V with respect to x and put dV/dx = 0

we get x = 12 and x = 4

x=12 is not possible so maximum volume is obtained when x=4 inch.