Part #1: Given a sheet of 8.5 x 11 paper, your group wants to construct an open topped box by cutting squares out of the corners. (Watch the demonstration.) You want to make a box with the greatest possible volume.

1. In your group, discuss a strategy for making a box, and write it in the space below. Be sure to explain why did you choose the dimensions you used.

2. Write the dimensions of your box below.

3. Find the volume of your box. Show your work

Part #2 Now we will study this problem algebraically.

1. Write a function for the volume of the box using the variable x = side of square you cut out to make the box.

2. Graph the function on your graphing calculator. Sketch the graph below, and include your window size. Use the graph to find the maximum volume.
   

3. Indicate the point on the graph that represents your box. How close was your prototype to the maximum volume?

Part #3 Now we will study the zeros of the function.

1. Estimate the zeros of the function using the trace function of your calculator.

2. What do the zeros tell us about our problem?

3. What portion of the domain is relevant to finding the maximum volume? Why?

Part #4 Now lets change the rules.

1. You want to construct a box with exactly ½ the maximum volume. What are the dimensions of this new box?  Show it on your graph.

2. Are these the only dimensions of such a box? Explain.