Question

PLEASE HELP!!!!!!

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.
What is the size of the cutout he needs to make so that he can fit the most marbles in the box?
If Thomas wants a volume of 12 cubic inches, what size does the cutout need to be? What would be the dimensions of this box?
Using complete sentences, explain the connection between the cutout and the volume of the box.
Design an equation that would work for any cardboard sheet length, q, and width, p.

Answer 1

Using a qxp sheet, the equation would be:

Vol = x(p-2x)(q-2x) where x is the height of the box sides.

If the Volume is 12 in^3, then
12 = (12-2x)(8-2x) = x(96 -40x + 4x^2) where x is the size of the cutout which would be the height of the box.

Graphing this on a TI Calculator where
The cubic function intercepts the line y=12,
yields x=3.65" as the cutoutsize of the box height.

If you've had calculus, you could take the derivative of the function and discover that the cutout needs to be 1.56 inches for the maximum volume.