See the attached picture.
- Create an equation for the volume of the box, find the zeroes, and sketch the graph of the function.
The resulting box has a volume
because the volume of a box is the product of its width 
, length 
, and height 
.
You know right away from the factored form of 
that the zeroes are 
. (zero product property)
- sketch the graph of the function
Easy to plot by hand. You know the zeroes, and you can check the sign of for any values of 
between these zeros to get an idea of what the graph of looks like. See the second attached picture.
Here's what I mean by "check the sign" in case you don't follow. We know 
when 
and 
. So we pick some value of between them, say 
, and find that
which is positive, so will be positive for any other between 0 and 8. Similarly we would find that 
for between 8 and 12, and so on.
- What is the size of the cutout he needs to make so that he can fit the most marbles in the box?
It's impossible to answer this without knowing the volume of each marble...
- If Thomas wants a volume of 12 cubic inches, what size does the cutout need to be?
Thomas wants 
, so you solve
While this is possible to do by hand, the procedure is tedious (look up "solving the cubic equation"). With a calculator, you'd find three approximate solutions
but you throw out the third solution because, realistically, the cutout length can't be greater than either of the sheet's dimensions.
- What would be the dimensions of this box?
The box's dimensions are ( in) x (
in) x (
in).
If , then 
and 
.
If , then 
and 
.
