Investigation: Maximizing area with a fixed perimeter

Scenario

Jill has a rambunctious puppy who needs a pen in her large property. There was a sale at the hardware store and she splurged and bought some fencing. Now she needs to determine how to best set up the fencing. Her only other restriction is that she only bought four corner poles, so it must be a rectangle.

Objectives

• To determine the maximum area of a rectangle given a fixed perimeter using technology
• To use math to think creatively in situations with context

Procedure

1. Suppose Jill bought 20 ft of fencing. With a partner, create a conjecture for what dimensions will give the maximum area for a perimeter of 20 ft. 
   The formulas P = 2 L + 2 W or the rearranged version W = \frac{P - 2 L}{2} might be helpful.
2. Double-check that the length and width you have predicted do indeed give a perimeter of 20 ft.
3. Use the GeoGebra animation test your conjecture.
   1. First set the Perimeter slider to 20 for the amount of fencing.
   2. Next set the Length slider to the length you think will give the biggest area. The width and area will be automatically calculated.
   3. See if you are correct by sliding the Length slider along to see if you can get a larger area with different dimensions.
   
   
   Created with Geogebra
4. With your partner, form a conjecture about, in general, what type of rectangle will give the greatest area.
5. Test your conjecture by using the GeoGebra animation above with different perimeters. Maybe try for a perimeter of 24 ft, 40 ft, and 60 ft.
6. Prove your conclusion by example. Complete the table below for 40 ft. The first row is done for you.

Discussion questions

1. In general, what type of rectangle gave the largest area?
2. If Jill bought 80 m of fencing, what rectangular dimensions should she use so that her playful puppy has the most space possible?
3. Consider that we often use the house as one of the sides for the fence. Would anything change if she only needed to fence three of the four sides of the rectangular play area? Suppose she had 20 m of fencing. What rectangular dimensions should give the maximum area? Use trial and error or another strategy to help you.
4. If she weren’t restricted to a rectangle, what other shapes might she consider for an even larger area with the restricted fencing? Use the GeoGebra applet below to help you by sliding the number of sides.