Calculus question

Question

A box is to be made out of a 8 by 14 piece of cardboard. Squares of equal size will be cut out of each corner, and then the ends and sides will be folded up to 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≤L ).

Answer 1

8 by 14

cut squares from corners,

what's the dimensions that max volume?

let d=x.w=8-2x, L=14-2x

V= dwL = x(8-2x)(14-2x)

V'=12x^2-88x+112x=0

use the quadratic formula

x=d=about 1.64 deep

w=about 4.72 wide

L=about 10.72 Long

Answer 2

Volume V = (14 - 2x)(8 - 2x)x = 4x(28 - 11x + x²) = 4x³ - 44x² + 112x; V' = 12x² - 88x + 112 = 0;

3x² - 22x + 28 = 0; x = (11 ±√121 - 84)/3; x = 11 ±√37)/3; x = (11 - √37)/3 = 1.64 units

V'' = 24x - 88; V''(1.64) = 24·1.64 - 88 = - 48.64 < 0. So, we have maximum.

L = 14 - 2·1.64 = 10.72 units; W = 8 - 2·1.64 = 4.72 units and H = x = 1.64 units