Question

A box without a top is to be made from a 30 cm by 30 cm rectangular piece of cardboard by cutting out square corners with a side length of x and then folding up and taping the sides.

What is the maximum possible volume of the box?

What value of x maximizes the volume of the box?

Answer 1

The dimensions of the box will be (30 -2x) cm square by x cm deep. The volume is the product of the length, width, and depth.
V = x(30 -2x)²
A graphing calculator can show the local maximum of this function.

a) The maximum possible volume is 2000 cm³.

b) The value of x that maximizes the volume of the box is 5 cm.