At first glance, this applet, ClosedBox2, contains many of the same components as the first Box Problem applet; however, the cut length determines the positioning of the cut so that in each case the box volume is relatively maximized.

Click Here to open the second Box Problem applet

**Warning**: The second Box Problem applet page, entitled ClosedBox2, is best viewed in 1024 x 768 resolution or greater and may take up to a minute to load.

In this applet there are a variety of elements that can be seen. First, the point **P** is no longer adjustable but is rather determined by the length of the cut defined by the segment BQ. In addition, the grey box in the lower left-hand of the applet contains a dynamic graphical depiction of the functional relationship between cut length and volume. That is, it contains a graphical depiction of

\[ V(l) = \left( {B - 2l} \right)\left( {{1 \over 2}A - 2l} \right)\left( {2l} \right) \]

where *l* corresponds to \( m( \overline {BQ} ) \) and when you intially open the applets, *B* = 8.5 and *A* = 14.0. The formula above should be equivalent to the formula you found in response to question #18.

Pick particular values for the length and width of the piece of cardboard, i.e. **A** and **B**, and then investigate the graph of the function on a hand-held graphing calculator. After doing so, answer the following questions:

- Why does the graphing calculator seem to show you more of the graph than the ClosedBox2 applet does?
- Why does the applet truncate the graph?
- What conditions should be on the domain of the function and how do they relate to physically constructing a box? Before answering this question, you might want to consider:
- Is the cut length constrained by the width of the piece of cardboard? If so, how and if not, why not?
- Is the cut length constrained by the length of the piece of cardboard? If so, how and if not, why not?

So far, these questions have focused primarily on a static rectangular sheet of cardboard. Now, you really need to think and explore to answer the following questions:

- For different lengths or widths, the applet's graph seems to change shape near the right-hand terminus, what mathematical reason can you provide for this change or provide an argument that it does, in fact, not change?
- Are there two (or more) non-isomorphic sheets of cardboard, so the maximal volume is the same? If so, identify them and if not, explain why not.
- Are there two (or more) non-isomorphic sheets of cardboard, so the placement of the maximal cut is the same? If so, identify them and if not, explain why not.