Box Folding Problem - Maximizing Volume[br][br]A piece of cardboard is rectangular. We cut a square that is [i]h[/i]" square from each corner and fold up the sides to form a box. What length should [i]h[/i] be to maximize the volume of the box?

A. What changes with the[color=#0a971e] Box Open/Close[/color] slider?[br][br]B. What changes with the [color=#0a971e]Scale[/color] slider? The [color=#d69210]Angle[/color] slider?[br][br]Change the [color=#1551b5]length[/color] of the box to 4" and the [color=#c51414]width[/color] to 6". Set the height to 1.5".[br][br]1. Look at the box. Is the actual width of the box 6"? If not, what does the width slider change?[br][br]2. How can we calculate the volume of the box? What is a formula we can use?[br][br]3. Use your formula to find the volume of a box that is 5" long, 5" wide and 0.5" tall.[br][br]4. Click the "Show Volume" box. Set the sliders to length = 5", width = 5" and h = 0.5". The volume is 8 inches cubed. Why does the length*width*height formula not work? Explain .[br][br]5. Set the length to 4" and the width to 6". Move the [i]h[/i] slider. What would a graph of the h values look like?[br][br]6. Click the "Show Volume Graph" box. Move the [i]h[/i] slider. Were you correct?[br][br]7. How can you use the yellow curve to find the largest volume of the box? What do the coordinates tell you?[br][br]8. Explain how to find a formula for the volume of a box cut from a sheet of paper that is 8"x16". Imagine you are explaining to a student who is not very good at math. He needs to know how big his [i]h[/i] squares will be to get maximum volume. Do not just give him a formula.[br][br]9. Now use the sliders to create the 8"x16" box. What value for [i]h[/i] creates the biggest box?