Equal squares are cut from the corners of a square sheet of side 24 cm and the sides folded up to make an open box. What side of square, in cm, gives the largest volume?

Explanation

1. 01

   Volume $V = x(24 - 2x)^2$ for $0 < x < 12$.

2. 02

   $V' = (24 - 2x)(24 - 6x) = 0$ gives $x = 4$ (the other root, 12, leaves no sheet).

3. 03

   At $x = 4$ cm the volume is $4 \times 16^2 = 1024$ cm³, the largest.