Pi Machine Week 2: Looking at the relation between radii and areas

Activity info on ESCOT web site.

Introduction

This week we will investigate the relation between the radius and area of circles. We will do this using another simulation. In this simulation, we have two concentric circles. The inner circle has a radius of 7 feet and the outer circle has a radius equal to twice the radius of the inner circle, or 14 feet. Points will be generated randomly anywhere in the two circles. Which of the following choices do you think best estimates the percent of points that will be in the inner circle?

1. 25%
2. 50%
3. 75%
4. 95%

Now think about any two concentric circles where you can change the radius of the inner circle and the outer circle will adjust so that it always has a radius that is double the size of the inner circle. For example, if you change the radius of the inner circle to 5, the outer circle will change to a radius of 10. If you change the radius of the inner and outer circles in this way, will the percentage of points falling in the inner circle change?

Please record your predictions:

1. The percentage of the randomly generated points will fall in the inner circle is ....... because .......

2. If you change the radius of the inner and outer circles in this way, the percentage of points falling in the inner circle (will/will not) change because......

Activity

Collect data to investigate your predictions.

One way to check your predictions is to run an experiment. You can use the computer simulation provided to do that.

We suggest that you experiment using several different numbers of points and several different sized radii, by following the instructions below.

To run the simulation, please follow these steps carefully and in order:

1. If you want to change the inner radius, select the Hand tool in the worksheet and click it on the little plus or minus agents on the right of the circle to increase or decrease the value of the inner radius. It should take values from 1-10. You can experiment with all the values in this range. The outer radius will always be twice the length the inner radius. Please note that you will not see the circles change size until you get to step 3.
2. Change the NumberOfPoints simulation property to reflect the number of points you want the simulation to generate. We suggest that you use 100, 1000, 10000, and 100000 points. Please note, that the more number of points, the more accurate your results. However, you should be aware that experiments with smaller radii and large number of points will take a long time to run.
3. Click the "Run" button.
4. When the simulation is done (the "Number of Points" will count down to 0, even though the "Total Points" will not change), record the results of your experiment.
5. Before running the simulation again (for different radii or number of points), you need to hit the "Reset" button. Wait for the picture in the Worksheet to completely reload and repeat steps 1 through 4.

Please anwer the following questions:

1. How do the results from the simulation compare with your predictions?

2. Write a mathematical explanation of what is going on? HINT: Consider the areas of each of the two circles.

3. Did the percentage of points falling inside the inner circle change as you changed the radius? Explain your results mathematically.

BONUS: In this simulation, the radius of the outer circle was 2 times the radius of the inner circle. Now consider a different situation: What percentage of points do you think would fall in the inner circle if you made the radius of the outer circle 3 times the radius of the inner circle? Please explain your answer.