An open top box with a square base is to be made from a square piece of cardboard that measures 24 inches on each side. A square will be cut from each corner of the cardboard and the sides will be turned up to form the box. Find the dimensions that will give the largest possible volume

4 inches deep by 16 inches square (the cut square is 4 inches)

Step-by-step explanation:

If x is the side of the square cut from each corner, the folded depth of the box is x, and the dimensions of the square base are 24-2x on each side. Then the volume is ...

V = x (24-2x) ^2

The derivative of volume with respect to x is ...

dV/dx = (24-2x) ^2 + x (2) (24-2x) (-2) = (24-2x) (24 - 6x)

Values of x that make this zero are 24/2 = 12 and 24/6 = 4. The latter value is the one of interest.

The box is 4 inches deep and 16 inches square.