## Double Ferris Wheel

On this web page you can explore the motion of a double Ferris wheel. This will help you solve Exercise 12 in Lesson 10.5 of Discovering Advanced Algebra: An Investigative Approach, and it might give you some ideas for how to create your own simulation using The Geometer's Sketchpad.

Each wheel has diameter 20 ft and takes 20 s to make a single rotation. The center of the lower wheel is 12 ft above the ground, and the center of the upper wheel is 34 ft above the ground. The center of the two-wheel set is 23 ft above the ground and takes 30 s to rotate once.

### Sketch

The sketch below shows a model of the double Ferris wheel. Press the buttons to rotate just the two wheels, just the two-wheel set, or both rotations. Press a button again to stop the motion. Press Start Over to return Sandra to the bottom position. (Note: Depending on your computer and web browser, the wheels may not rotate at exactly the correct speeds.)

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### Investigate

1. Sandra gets on at the foot of the bottom wheel. Press Rotate Small Wheels and watch Sandra's height above the center of the bottom wheel as the wheels rotate. Sketch a graph and write an equation that models her height above the center of this wheel as a function of time.
2. The entire ride (the two-wheel set) starts revolving at the same time that the two smaller wheels begin to rotate. Stop the small wheels, press Rotate Two-Wheel Set, and watch the height of the center of Sandra's wheel above the ground. Sketch a graph and write an equation that models the height of the center of Sandra's wheel as a function of time.
3. Stop the two-wheel set, press Start Over to return Sandra to the bottom, then press Both Rotations. Watch Sandra's height above the ground. Sketch a graph of her height above the ground as a function of time.
4. Because the two motions occur simultaneously, you can add the two equations (from questions 1 and 2) to write a final equation for Sandra's position. Write this equation. Graph the equation on your calculator and compare it to the sketch you made.
5. During a 5 min ride, for how many distinct time periods is Sandra within 6 ft of the ground? How can you use the sketch to answer this question? How can you use the graph to answer this question?

### Sketch

This sketch allows you to trace Sandra's position above the ground as a function of time. You can rotate just the small wheels, just the two-wheel set, or both rotations. To start over, click a button again to stop the motion, press Start Over to return Sandra to the bottom position, and then click X to erase the traces.

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### Investigate

1. Does each graph agree with your results from the first part of this exploration? If any of the graphs are different, explain why they are different.
2. After how many seconds does the motion of the entire ride (Both Rotations) repeat itself? How is this shown on the graph?
3. Trace Both Rotations to find how many distinct time periods Sandra is within 6 ft of the ground during a 2 min ride. How can you use recursive thinking to extend your answer to 5 min? 10 min?
4. How would Sandra's position above the ground change if her small wheel is first rotated to let other people on the ride? Start over and press Rotate Small Wheels to move Sandra to a different position, then press Rotate Two-Wheel Set to move the entire ride. How does the graph compare to the graph of when Sandra started at the bottom? Discuss any similarities or differences between the two graphs.