Making the Most of Your Space

You are the manager of a band that has been hired to perform at a party, and you want to create the largest dance floor possible for the attendees.  If you only have a given length of rope to delineate the space, how can you make the most of your space?

Previous knowledge: Students should be able to write the equation of a quadratic function in vertex form.

Objective:  At the completion of this lab, students will be able to write the equation of a parabola to model the length (or width) versus area of a specified perimeter.

Materials:  graph paper and graphing calculator.

Group Size: 2

Procedure:
1. Each group will be given a specific length for their rope (perimeter). For the lab to work easily the teacher should assign each group a specific perimeter that is a multiple of four.

2. Set up a table as follows, including all possible whole number lengths and widths (including 0) and find the corresponding area:
 Length Width Area 0 0 1 1 0 0
3. On your graph paper, create a scatter plot using length as the independent variable and area as the dependent variable.  Make sure you choose an appropriate interval and label for each axis.

4. Using the vertex of your parabola, write the equation of the quadratic function that fits your scatter plot in vertex form.

5. Rewrite the equation from #4 in standard form.  Make sure you show all of your work.

6. Enter the Lengths from your table into List 1 and their corresponding Areas into List 2.  Give the window you set to see the entire scatter plot:
Xmin ______ Xmax ______ Xscl ______

Ymin ______ Ymax ______ Yscl ______
7. Check that the equations you wrote are correct by putting the vertex form of the equation in Y1 and the standard form of the equation in Y2 (in the Y= screen) and graph them.  Do you get the same parabola?  Do they fit your data?  If not, you need to go back and fix your work!

8. Each group should display their data in a class chart in the front of the room.

9. Which length gave you the maximum area?  How does this length relate to your perimeter?

10. On your TI-83, put the widths in L3.  Make a new scatter plot using the widths as your independent variable and the corresponding areas as the dependent variable.

11. How do the equations you wrote in #4 and #5 fit this data?  Why?

12. What width gave you the maximum area?  How does this relate to the length that gave you the maximum area?

13. A class discussion should take place of the length and width that yielded the maximum area for each group and where that data piece was located on the graph.

14. Why did we say that the easiest perimeter to work with was one that was a multiple of four?
Homework: Make sure that you show your work.
1. Pick a perimeter that is not a multiple of 4.

2. Make a table like you did in #2 using integral lengths for your length and width.

3. On your graph paper, make a scatter plot with the length as the independent variable and the area as the dependent variable.

4. Draw the parabola that fits the data on your graph.

5. What is the vertex of this parabola? Is it one of your data points? Why or why not?

6. Write the equation of the parabola in both vertex and standard form:

7. Find the length and width that would give you the maximum area:

8. What is the optimum shape that yields the largest area for any given perimeter?