George is trying to create a rectangular box with a square base to ship his product in. he wants it to enclose the maximum volume but he only has 56 square feet of material to use. in order to find the dimensions he sets up two equations:

To solve the problem we must set the equation for the volume and surface area

let x be the side of the square base

and h be the height of the box

so the volume, v = hx^2

and the area, a = x^2 + xh = 56

then solve for the equation of h

x^2 + xh = 56

xh = 56 - x^2

h = 56/x - x

to solve the maximum volume, solve the first derivative of the volume and equate to zero

v = hx^2

v = (56/x - x) x^2

v = 56x - x^3

dv/dx = 56 - 3x^2

0 = 56 - 3x^2

3x^2 = 56

x = 4.32 ft

h = 8.64 ft