Thomas buys a cardboard sheet that is 8 by 12 inches. Let x be the side length of each cutout. Create an equation for the volume of the box, find the zeroes, and sketch the graph of the function.

Volume of the box = area of the base * height

Side of each cutout = x

Length of the base = 12 - x - x = 12 - 2x

width of the base = 8 - x - x = 8 - 2x

area of the base = (12 - 2x) (8 - 2x) = 12*8 - 12*2x - 8*2x + 4x^2 = 96 - 40x + 4x^2

height = x

Volume = (96 - 40x + 4x^2) x = 96x - 40x^2 + 4x^3

Equation of the volume of the box = 96x - 40x^2 + 4x^3

Zeros of the function: use the factored form:

x (12 - 2x) (8 - 2x) = 0

=> x = 0, x = 6 and x = 4

Sketch of the graph:

The graph comes growing from ( - infinity, - infinity), crosses the origin (0,0), grows until a local maximum before 2, starts to decrease, intercepts the x axis at x = 4, continues decreasing until a local minium before 6, starts to increase again, crosses the x axis at x = 6, and continues increasing toward infinity. If you are using derivatives, you can find the local minimum and maximum.