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:
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: