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:
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: