## MATH100: Maximizing Box Volume

The purpose of this exercise is to use the graph of a third degree polynomial function to find the maximum possible volume of a box for shipping. This exercise is based on problem 104, p. 160.
A delivery service refuses to deliver any package for which the length plus the girth is more than 120 inches. You wish to design a box with square cross-section that will hold the maximum possible volume and can be shipped by this service.
1. Find a partner to work with; either you or your partner should have a laptop computer with Graphmatica installed.
2. Copy the diagram of the box from your text in the space below.

3. Use the diagram to write an equation for the volume of the box.

4. Use the diagram to write an equation for the girth of the box.

5. According to the diagram, what is the length of the box?

6. Since you want to maximize the volume, assume that the length plus the girth is as large as possible: 120 inches. Write an equation for this.

7. Check your answer to the previous question and your equation for the volume of the box by comparing them to another group's answer. Correct any mistakes before moving on to the next step.
8. In order to maximize the volume of the box, you would like to graph a volume function and then find the maximum output of that function. Explain why you can't graph the volume function you found earlier.

9. If you could describe the volume of the box in terms of only one variable, you could then graph the volume using Graphmatica. Use your equation setting length plus girth equal to 120 to solve for y in terms of x.

10. You can now replace y in your equation for the volume of the box by the formula you got in the previous question. Write the resulting function below; check that it matches the solution given in part (a) in your text.

11. Graph this function using Graphmatica. For what value of x do you get the biggest box volume?

12. If x has the value found above, what is the value of y?

13. What is the maximum possible box volume, assuming a square cross-section?

14. Check your answers to the preceding three questions against those of another group.
15. Look at your graph of the volume of the box and determine for what values of x the volume equals 13,500 cubic inches. (This is equivalent to finding the zeros of the polynomial function V(x) - 13,500.)

16. Can you actually build a box that corresponds to each of these values of x? If not, why not?