A square of size x inches is cut out of each corner of an 8in by 12in piece of cardboard, and the sides are folded up to form an open-topped box. Determine the dimensions of the cut-out squares that will produce the box of maximum volume.

Answer 1

Volume = x(8-2x)(12-2x) , where 0 < x < 4
= 96x -40x^2 + 4x^3
d(volume)/dx = 96 - 80x + 12x^2
= 0 for a max/min of volume
3x^2 - 20x + 24 = 0

use the quadratic formula to solve, reject the x value outside our stated domain above