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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.

David Meel and Thomas Hern, "Student Module for 'Thinking Outside the Box'," *Loci* (August 2009)