Concept Byte: Comparing Perimeters and Areas

Use With Lesson 1-8

TECHNOLOGY

You can use a graphing calculator or spreadsheet software to find maximum and minimum values. These values help you solve real-world problems where you want to minimize or maximize a quantity such as cost or time. In this Activity, you will find minimum and maximum values for area and perimeter problems.

Activity

You have 32 yd of fencing. You want to make a rectangular horse pen with maximum area.

1. Draw some possible rectangular pens and find their areas. Use the examples below as models.

   

2. You plan to use all of your fencing. Let X represent the base of the pen. What is the height of the pen in terms of X? What is the area of the pen in terms of X?
3. Make a graphing calculator table to find area. Again, let X represent the base. For Y 1 , cap y sub 1 , comma  enter the expression you wrote for the height in Question 2. For Y 2 , cap y sub 2 , comma  enter the expression you wrote for the area in Question 2. Set the table so that X starts at 4 and changes by 1. Scroll down the table.
   1. What value of X gives you the maximum area?
   2. What is the maximum area?
4. Use your calculator to graph Y 2 . cap y sub 2 , .  Describe the shape of the graph. Trace on the graph to find the coordinates of the highest point. What is the relationship, if any, between the coordinates of the highest point on the graph and your answers to Question 3? Explain.

Exercises

5. For a fixed perimeter, what rectangular shape will result in a maximum area?
6. Consider that the pen is not limited to polygon shapes. What is the area of a circular pen with circumference 32 yd? How does this result compare to the maximum area you found in the Activity?
7. You plan to make a rectangular garden with an area of 900   ft 2 . 900 , ft squared , .  You want to use a minimum amount of fencing to keep the cost low.
   1. List some possible dimensions for the garden. Find the perimeter of each.
   2. Make a graphing calculator table. Use integer values of the base b, and the corresponding values of the height h, to find values for P, the perimeter. What dimensions will give you a garden with the minimum perimeter?

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