Application of a 3rd degree polynomial

Question

A piece of cardboard measuring 11 inches by 10 inches is formed into an open-top box by cutting squares with side length x from each corner and folding up the sides.

Find a formula for the volume of the box in terms of x

Find the value for x that will maximize the volume of the box

Answer 1

Length = 11 - 2x

width = 10 -2x

height is x

V = (11-2x)(10-2x)x

= (110 - 22x - 20x + 4x^2)x

= 110x - 42x^2 + 4x^3

factoring out the x the resulting quadratic is 4x^2 - 42x + 110

which maximizes at -b/(2a) = 42/8 = 21/4 = 5.25

is an APPROXIMATE answer

using calculus, 0 = dV/dx = 12x^2 - 84x + 110

6x^2 - 42x + 55

which per quadratic formula has solution 5.25594......

if you don;t like the calculus, examine hte graph of the cubic polynomial

and find where it maxes