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 [math]x[/math] from the corners, then folding the "flaps" up to form sides. (The top stays open).[br][br]Leave the two check boxes unchecked to begin.[br][br]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 "[math]x[/math]" 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.[br][br]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 [math]x[/math], so we might expect the volume to depend on [math]x[/math] 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 [math]x[/math]? If so, simply multiply them together to get [math]V(x)[/math], the Volume function. Check the "Show Vol Calculation" box to check your answer.[br][br]Finding the exact value of [math]x[/math] that maximizes the volume requires calculus. However, you can make a good estimate by finding the maximum value on the graph of [math]V(x)[/math]. You can display the graph by checking the "Show Vol Graph" box. The actual value of [math]V[/math] for the current value of [math]x[/math] is plotted as a point, and the "optimum" value of [math]x[/math] (the value that gives the maximum volume) is plotted on the [math]x[/math]-axis.[br][br]Think about this question: Part of [math]V(x)[/math] 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 [math]x[/math] 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.