Looking at the relation between radii and areas

Make a Prediction.

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?

]]> 7.0 100.0 0.0 0.0 0.0 0.0 Collect data to investigate your predictions.

One way to check your predictions is to run an experiment. You can use the computer simulation provided in this activity page 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 do not see two circles in the "Worksheet" area on the right, hit the "Reset" button and wait until they show up. It may take some time to reload the picture in the simulation window, so please be patient!
2. If you want to change the inner radius, move the slider bar under the simulation to the number you wish. The inner radius can take values from 1 to 10. You can experiment with all the values in this range. The outer radius will always be twice the length the inner radius. You will not see the circles change size until you get to step 4.
3. Click the box called "Total Points" and enter 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.
4. Click the "Run" button (located under the simulation).
5. When the simulation is done (the "Points Left" will count down to 0, even though the "Total Points" will not change), click on the "Sample" button (located next to the table). Notice how the chart shows the results of your experiment.
6. 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" area to completely reload and repeat steps 2 through 6.

When you are done experimenting, answer the activity questions in the answer box below.

Note: You should close the ESCOT Browser when you finish this activity. If you want to run other activities, run the Browser again.

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