Word problem

Question

an open-lid is being made from a 12 inch by 14 inch piece of cardboard by cutting out x in corners. What is the height of the box that will give the greatest volume?

Answer 1

V = (12-2x)(14-2x)x = 4x(6-x)(7-x) = 4x(42-13x+x^2 = 168x-52x^2 + 4x^3

V' = 168 - 104x + 12x^2 = 0

42-26x + 3x^3 = 0

x= 26/84 + (1/84)sqr(26^2 -4(42(3))

= 13/42 + (1/84)sqr372

= 13/42 +19.3/84 = 45.3/84 = about 0.54 inches for height that gives maximum volume

half an inch would give volume 11(13)(1/2 = 71.5 cubic inches