# Ferris Wheels—Tracking the Height of a Passenger Car

### Ferris Wheels—Tracking the Height of a Passenger Car

Student Outcomes

• Students apply geometric concepts in modeling situations. Specifically, they find distances between points of a circle and a given line to represent the height above the ground of a passenger car on a Ferris wheel as it is rotated a number of degrees about the origin from an initial reference point.
• Students sketch the graph of a nonlinear relationship between variables

### New York State Common Core Math Algebra II, Module 2, Lesson 1

Worksheets for Algebra II, Module 2, Lesson 1

Solutions for Algebra II, Module 2, Lesson 1

Classwork

Exploratory Challenge 1: The Height of a Ferris Wheel Car George Ferris built the first Ferris wheel in 1893 for the World’s Columbian Exhibition in Chicago. It had 30 passenger cars, was 264 feet tall and rotated once every 9 minutes when all the cars were loaded. The ride cost \$0.50.
a. Create a sketch of the height of a passenger car on the original Ferris wheel as that car rotates around the wheel 4 times. List any assumptions that you are making as you create your model.
b. What type of function would best model this situation?

Exercises 1–5

1. Suppose a Ferris wheel has a diameter of 150 feet. From your viewpoint, the Ferris wheel is rotating counterclockwise. We will refer to a rotation through a full 360° as a turn.
a. Create a sketch of the height of a car that starts at the bottom of the wheel and continues for two turns.
b. Explain how the features of your graph relate to this situation.
2. Suppose a Ferris wheel has a diameter of 150 feet. From your viewpoint, the Ferris wheel is rotating counterclockwise.
a. Your friends board the Ferris wheel, and the ride continues boarding passengers. Their car is in the three o’clock position when the ride begins. Create a sketch of the height of your friends’ car for two turns
b. Explain how the features of your graph relate to this situation.
3. How would your sketch change if the diameter of the wheel changed?
4. If you translated the sketch of your graph down by the radius of the wheel, what would the 𝑥-axis represent in this situation?
5. How could we create a more precise sketch?

Exploratory Challenge 2: The Paper Plate Model

Use a paper plate mounted on a sheet of paper to model a Ferris wheel, where the lower edge of the paper represents the ground. Use a ruler and protractor to measure the height of a Ferris wheel car above the ground for various amounts of rotation. Suppose that your friends board the Ferris wheel near the end of the boarding period and the ride begins when their car is in the three o’clock position as shown.
a. Mark the diagram below to estimate the location of the Ferris wheel passenger car every 15 degrees. The point on the circle below represents the passenger car in the 3 o’clock position. Since this is the beginning of the ride, consider this position to be the result of rotating by 0°.
b. Using the physical model you created with your group, record your measurements in the table, and then graph the ordered pairs (rotation, height) on the coordinate grid shown below. Provide appropriate labels on the axes
c. Explain how the features of your graph relate to the paper plate model you created. Closing

• How does a function like the one that represents the height of a passenger car on a Ferris wheel differ from other types of functions you have studied such as linear, polynomial, and exponential functions? What is the domain of your Ferris wheel height function? What is the range?
• Provide a definition of periodic function in your own words. Why is the Ferris wheel height function an example of a periodic function?
• What other situations might be modeled by a periodic function?

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