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

A. What changes with the Box Open/Close slider?
B. What changes with the Scale slider? The Angle slider?
Change the length of the box to 4" and the width to 6". Set the height to 1.5".
1. Look at the box. Is the actual width of the box 6"? If not, what does the width slider change?
2. How can we calculate the volume of the box? What is a formula we can use?
3. Use your formula to find the volume of a box that is 5" long, 5" wide and 0.5" tall.
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 .
5. Set the length to 4" and the width to 6". Move the h slider. What would a graph of the h values look like?
6. Click the "Show Volume Graph" box. Move the h slider. Were you correct?
7. How can you use the yellow curve to find the largest volume of the box? What do the coordinates tell you?
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 h squares will be to get maximum volume. Do not just give him a formula.
9. Now use the sliders to create the 8"x16" box. What value for h creates the biggest box?