Dimensions of the bottom of the box: x Height of the box:

Question

An open box is to be made out of a 12-inch by 18-inch piece of cardboard by cutting out squares of equal size from the four corners and bending up the sides. Find the dimensions of the resulting box that has the largest volume.

Answer 1

V = x(12-2x)(18-2x) = x(216 - 60x + 4x^2) = 216x - 60x^2 + 4x^3 = V(x)

dV/dx = 216 - 120x + 8x^2 = 0

divides by 8 and re-arranges:

x^2 -15x + 27 = 0

(x-7)(x-8) = 0

x=7 or x=8

plug in to find V(7) and V(8) for max volume