Max volume of a box

Question

A sheet of cardboard 18 inches square is to be used to make an open-top box by cutting squares of equal size from the corners and folding up the sides. What size squares should be cut to obtain a box with maximum volume?

Answer 1

The bottom of the box is 18-2x by 18-2x with a height of x

V = x(18 - 2x)²

V = x(324 - 72x + 4x²)

V = 4x³ - 72x² + 324x

 

Maximum occurs when V' = 0.