The task: To create the lidless box with maximum volume by removing a square from each corner of a 5-inch by 8-inch sheet of paper.

Click on the animation button in this GSP file to look at how the volume of the box changes as the side length of the square removed changes. What is the longest possible side of the square removed? At what side length does the volume seem to be maximized?

To get a sense of how the box's volume varies with the size of the corner piece removed, let's start by looking at some data. Below is a chart showing the side length of the piece removed and the resulting box length, width, height, and volume. Does the relationship seem linear (steadily increasing or decreasing)? Or do things seem a bit more complex?

From these data we see a maximum volume of
18.00in^3 when a square with side length of 1 inch is removed
from each corner. Hmmm . . . since our data are discrete (that
is, not continuous) how do we know that removing a 1-inch square
results in a box with the maximum volume ? Perhaps removing 1.1-in
squares create a box with greater volume. Let's zoom in a bit
to see the volumes of boxes with side lengths *near* 1 inch.

Well, our theory that the greatest volume possible occurs when you remove a 1 inch square is supported. But, we again run into the problem of our data being discrete - we just can't see what's happening between 0.95 and 1.05 and there is always a chance that boxes having a volume greater than 18.0in^3 happen when the squares have a side length between this range.

So, it seems like we need a continuous (and
not discrete) picture of how volume is changing with side length.
To find this picture, we'll turn the functions. Basically we want
to find function that describes the volume of the lidless box
for *any* side length of the square corner removed. Since
the volume is the product of the lenght, width, and height all
we need to do is figure out how to describe each of these in terms
of the side length of the removed square. Let's look back at our
template to figure this out.

**Side length of square removed:** Since this is what we'll vary, let's call the side
length '**X**'

**Length of the Box:**
From the template we can see that the length of the box is related
to the 5-inch side of the piece of paper. In fact, the length
seems to equal 5 inches minus two side lengths, or **5-2X**.

**Width of the Box**:
From the figure we see that the width of the box is related to
the 8-inch side of the paper. Much like before, the width is 8
inches minus two side lengths, or **8-2X**.

**Height of the Box**:
If we imagine folding the template to form a lidless box, we see
that the height of the box is precisely the side length of the
square removed, or **X**.

**Volume of the Box**:
Using the knowledge that Volume = Length*Width*Height we find
that the volume of a box is the following:

So, we've found a function to describe how the volume of a lidless box varies with the side length of a square removed frome each corner. Let's take a look at a graph of this function.

Now we can see the cubic function that describes our problem. But a question remains, are we looking at the entire domain of this function, that is can we remove any side length, 'X', that is a real number? In our problem situation, the smallest length of side removed is 0 inches and the largest side length of the square removed is 2.5 inches (since the length of the box is 5-2X, any X greater than 2.5 will create a negative length). So in reality, we should only examine the portion of the graph where . Within this domain, we can clearly see that a maximum volume exists somewhere around X=1. You can click the 'Animate' button in this GSP file to see how both the volume graph and the shape of our box change as the side length 'X' changes.

To really find the exact side length where the volume of the lidless box is maximized, we turn to some calculus. Remember that when a function has a local maximum or minimum the derivative of the function at that point is 0 (or, the slope of the tangent line at the maximum/minimum is 0). So, to find our max, we just need to find the derivative of our function and solve for X when that derivative is 0. Here goes . . .

Now if we take a moment to reflect on our solutions for X, we see that 10/3 is outside of our domain (10/3>2.5). Therefore x=1 is the only solution for where our volume function reaches a maximum for . Tah Dah! Our hypothesis that removing squares with side length of 1 inch result in a maximum volume (18.00 inches cubed) is confirmed.

Well, we've looked our box's volume. What about it's surface area . . .?