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One element in the applets that we have designed are the multiple representations and alternative ways of conveying information. For instance, the applet, ClosedBox, allows you to explore various scenarios. You can manipulate the lengths A and B by just pulling on the yellow points **A'** and **B'**. The yellow points **P** and **Q** also move. In doing so, the other components of the applet change in accordance with these manipulations. The goal here is for you to interact with a wide variety of examples and see if the conjectures they make hold up to empirical investigation. In addition, you can move the corners of the box, in the lower right-hand corner of the applet, and see if the box will actually close or not, an important aspect if you want the box to hold something. The last elements in this applet are the two different graphical indicators of maximal volume. The one graph shows the volume with respect to **P** or **Q** while the other is held constant and the bar graph next to it displays the percentage of maximal volume obtained by the current configuration. If the volume is too large, one can resize the vertical unit, a yellow point denoted by **U**, to get the graphs comfortably into the grey viewing window.

Click Here to open the first Box Problem applet

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

You should consider the following questions as you are playing with the applet:

- Is there a relationship between the cut length and the position of the cut that maximizes the volume?
- Under what conditions does the volume drop to zero?
- Is there a best way to orient
**P**and**Q**so that the maximum is achieved? - How do the two graphics on the bottom left-hand side interact with each other?
- Why, under certain conditions, is the blue graph not connected nicely? Describe those conditions to the best of your ability.
- Why does one of the folds need to occur at the midpoint?
- What happens if the two folds aren't symmetric about the midpoint fold?

In the following table, carefully record the data from their various trials where the cut position, \( m( \overline {AP}) \) , is set and the cut length, \( m( \overline {BQ}) \) , is changed to maximize the volume for that particular cut position. The last row of each Trial is for you to search for the maximal cut length and the corresponding cut position. The final trial is for an open exploration of a piece of cardboard of the student's choosing.

\( m( \overline {BQ}) \) | \( m( \overline {AP}) \) | \( m( \overline {PM}) \) | \( m( \overline {QQ'}) \) | \( m( \overline {BB'}) \) | \( m( \overline {AA'}) \) | % of max volume | |
---|---|---|---|---|---|---|---|

Trial 1.1 | 1.5 | 3.0 | 4.0 | 5.5 | 8.5 | 14.0 | 97.75% |

Trial 1.2 | 3.5 | 3.5 | 8.5 | 14.0 | |||

Trial 1.3 | 4.5 | 2.5 | 8.5 | 14.0 | |||

Trial 1.4 | 8.5 | 14.0 | |||||

Trial 2.1 | 2.5 | 2.5 | 10.0 | 10.0 | |||

Trial 2.2 | 1.5 | 3.5 | 10.0 | 10.0 | |||

Trial 2.3 | 4.25 | 0.75 | 10.0 | 10.0 | |||

Trial 2.4 | 10.0 | 10.0 | |||||

Trial 3.1 | 3.0 | 2.5 | 8.5 | 11.0 | |||

Trial 3.2 | 2.75 | 2.75 | 8.5 | 11.0 | |||

Trial 3.3 | 4.0 | 1.5 | 8.5 | 11.0 | |||

Trial 3.4 | 8.5 | 11.0 | |||||

Trial 4.1 | |||||||

Trial 4.2 | |||||||

Trial 4.3 | |||||||

Trial 4.4 |

Exploring various positions of a cut, should lead you to a conjecture about the position of a cut and the cut length necessary to obtain the maximum volume.

Answer the following questions:

- For any positioning of the fold, the maximum volume can be best achieved when the cut length is set to?
- If the cut length exceeds the minimum of the length and width of the box, then what happens and why?
- If the cut length is less than half of the minimum of the length and width of the box, then what happens and why?
- Under what conditions is the blue graph connected above the
*x*-axis? - Under those conditions and for a particular length and width, what are the dimensions of the box with maximum volume?
- Why does the red graph always appear as a slanted portion? What mathematical meaning does this red graph have? Does this slanted portion ever change slope? If so, why and if not, why not?
- Using the results of your investigations and the variable
*l*to correspond to the cut length, determine a functional description of volume taking into account the length and width of the cardboard,**A**and**B**respectively, and the variable*l*. Before answering this question, you might want to consider:- How does one determine the volume of a rectangular box?
- What would be a description for the length of the box?
- What would be a description for the width of the box?
- What would be a description for the height of the box?
- How can you use these descriptions to build a functional relationship between volume and cut length?

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