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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.

Figure 9: The second Box Problem applet

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 currently *B* = 8.5 and *A* = 14.0.

One element that students need to grapple with when working with this particular applet is the graphical depiction of the function. In particular, graphing \( V(l) \) using a graphing calculator or computer algebra system yields a figure similar to the following:

Figure 10: Graph of the Box Problem Function

The graphical depiction of the function presented in the applet is a truncated version of the one in figure 10. Students will need to come to grips with the fact that the applet is only concerned with the volume of boxes that are physically constructible whereas the graph of the function shown in figure 10, does not necessarily concern itself with the constructability of the box. Instead, it provides a graph of the functional relationship between an independent variable *l* and a dependent variable *V*. In essence, this graph in figure 10 is less concerned with cut length and volume and more concerned with expressing the relationship for all possible values of *l*, irrespective if these values are possible cut lengths or if those cut lengths yield appropriate volumes.

Too often, we see students focused on the algebraic elements of a problem without considering the physical (or mathematical) constraints on that problem. We seek in this applet to guide students to grapple with the interplay of these two seemingly disparate forces. In turn, we hope to lead them to reconcile for themselves how functional relationships that model real-world phenomena require a careful examination of the domain for which that relationship actually does model the phenomena. For instance, students might at first think that *l*'s only restriction is that it must be less than \( {1 \over 2}m( {\overline {BB'} } ) \) since a cut cannot exceed half the width of the cardboard and maintain its connectedness. However, there is another constraint: the cut length, *l*, cannot exceed \( {1 \over 4}m( {\overline {AA'} } ) \) . This ''hidden'' constraint comes directly from the relationship of cut length and position of the cut and depends on the relationship between length and width of the rectangular piece of cardboard.

Thomas Hern (Bowling Green State Univ.) and David Meel (Bowling Green State Univ.), "Thinking Outside the Box -- or Maybe Just About the Box," *Loci* (February 2010), DOI:10.4169/loci003321