A box with an open top is to be constructed from a square piece of cardboard, 24 inches wide, by cutting out a square from each of the four corners and bending up the sides. Find the largest volume that such a box can have.

Answer 1

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

V = x(24-2x)^2 = 4x^3 - 96x^2 + 576x , where x < 12
dV/dx = 12x^2 - 192x + 576 = 0 for a min of V
x^2 - 16x + 48 = 0
(x-4)(x-12) = 0
x = 4 or x - 12, but x 12 , so x = 4

largest volume = x(24-2x)^2
= 4(16)^2 = 1024 inches^2

Answer 2

1728 inches

Answer 3

Why are you posting "1728 inches"?
What is that supposed to mean?