A square piece of cardboard, 24 inches by 24 inches, is to be made into an open box by cutting out each of the four corners. Each side of the box will then be folded up. Find the maximum volume that the box can hold.

Answer 1

let the size of the cut-out be x by x inches

length of base = 24-2x
width of base = 24-2x
height = x

V = x(24-x)^2
= x(576 - 48x + x^2)
=576x - 48x^2 + x^2
dV/dx = 576 - 96x + 3x^2 = 0 for a max of V
x^2 - 32x + 192 = 0
(x - 24)(x - 8) = 0
x - 8 or x = 24, but if x = 24, we have a negative side

so x = 8
V = 8(24-16)^2 = 512 inches^3