1. Minimizing Perimeter

2. The Problem A spectator viewing area for a downhill ski race needs to be fenced in. The organizers know they need 64 m 2 of enclosed space inside of the fenced area. They want to minimize the amount of fencing that they need so as not to create waste. What are the dimensions of the space with the smallest amount of fencing?

3. The Situation Consider a spectator viewing area of 64 m 2. We must examine this situation in order to determine the best shape to minimize the amount of fencing needed to surround the viewing area. Consider 3 possible configurations below. 64 m 2

4. The Situation Recall that the area of a rectangle is calculated by multiplying the length of the figure by it’s width. We can try integers first, so we need to think of factors of 64. Factors are two numbers that multiply to give 64.

5. The Situation These factors of 64 will be the dimensions of the rectangular area. Complete the chart that you opened from the main course page.

6. AreaFactorsLengthWidthPerimeter 64 m 2 1  64 1 m64 m130 m 64 m 2

7. The Solution Looking at your completed chart, what is the smallest perimeter? What dimensions give you the smallest perimeter? What is the shape of the figure that has the smallest perimeter?

8. The Explanation The smallest perimeter is achieved when the dimensions of the viewing area are a 8 m  8 m square. Therefore, the best possible design for the spectator viewing area is a square. So, just as when maximizing area, the best design for a minimum perimeter of any given area is a square.