A box is to be made out of a 12 by 20 piece of cardboard Squ

A box is to be made out of a 12 by 20 piece of cardboard. Squares of equal size will be cut out of each corner, and then the ends and sides will be folded up form a box with an open top. Find the length L, width W, and height H of the resulting box that maximizes the volume. (Assume that W le L). L= W= H=

Solution

length of cardboard = 20

width of cardboard = 12

let squares of size x is cut from each corner

then new length of box = 20 - 2x

width of box = 12 - 2x

and height = x

volume = length * width * height

volume = (20-2x)(12-2x)(x) = (240 - 24x -40x + 4x^2 )x

vloume = 4x^3 - 64x^2 + 240x

12x^2 - 128x + 240 = 0

to maximize volume x would be

x = height = 2.427

length = 20 - 2(2.427) = 15.14

width = 12 - 2(2.427) = 7.14