Question

You construct an open-top box by cutting equal sized squares (x by x inches) out of the four corners of a 14 inch by 24 inch sheet of metal and then folding up the sides. Find a function for the volume of the box as a function of x and then determine the largest volume possible.

Answer 1

Volume Box = l*w*h

l=24-2x

w = 14-2x

h = x

(24-2x)(14-2x)(x)

(24-2x)(14x-2x^2)

336x -48x^2 -28x^2 +4x^3

V = 4x^3 -76x^2 + 336x find the max vloume; get the derivative..

V' = 12x^2 -152x +336

v'=4(3x^2 -38x +84) find solve for V'=0

3x^2 -38x +84 = 0

38/6 +- sqrt(1444 - 1008)/6

19/3 +- sqrt(109)/3

x=9.8 or 2.85 is what I get if I didnt mess it up :)