Box - Folding for Maximum Volume

Topic:
Volume
How can we make a box with maximum volume from a given size "blank" of material? (This is a modification of Dani Novak's excellent demo...he did all the heavy lifting. All I did was move stuff around!)
You are given a "blank" of cardboard and asked to make a box out of it, that has the most possible volume. You make the box by cutting squares of side from the corners, then folding the "flaps" up to form sides. (The top stays open). Leave the two check boxes unchecked to begin. Set the desired size of your blank using the "Blank Length" and "Blank Width" sliders. Then try different sizes of square cutouts by changing the "" slider. Fold and unfold the side flaps using the "Box Open and Close" slider. If necessary, adjust "Angle" and "Zoom" to get the best view of the box in the right-hand viewing window. Remember that the volume of a rectangular prism is calculated as V = L W H. Notice that each of the three dimensions of the box will change with , so we might expect the volume to depend on as well. L is NOT the blank's length; it is the length of the box after the corners have been removed and the sides folded up. Same with H. Can you write a formula for each of L, H, and W in terms of ? If so, simply multiply them together to get , the Volume function. Check the "Show Vol Calculation" box to check your answer. Finding the exact value of that maximizes the volume requires calculus. However, you can make a good estimate by finding the maximum value on the graph of . You can display the graph by checking the "Show Vol Graph" box. The actual value of for the current value of is plotted as a point, and the "optimum" value of (the value that gives the maximum volume) is plotted on the -axis. Think about this question: Part of is highlighted in red; the rest is displayed as a gray dashed curve. What is the significance of the highlighting? Think about what the possible values of could be in the "real world". This is called "domain restriction". Often times, our mathematical models realistically apply only for a certain sub-domain of the function we are using.