# The Height and Co-Height Functions of a Ferris Wheel

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

Student Outcomes

• Students model and graph two functions given by the location of a passenger car on a Ferris wheel as it is rotated a number of degrees about the origin from an initial reference position.

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

Worksheets for Algebra II, Module 2, Lesson 2

Solutions for Algebra II, Module 2, Lesson 2

Classwork

Opening Exercise
Suppose a Ferris wheel has a radius of 50 feet. We will measure the height of a passenger car that starts in the 3 o’clock position with respect to the horizontal line through the center of the wheel. That is, we consider the height of the passenger car at the outset of the problem (that is, after a 0° rotation) to be 0 feet.
a. Mark the diagram to show the position of a passenger car at 30-degree intervals as it rotates counterclockwise around the Ferris wheel.
b. Sketch the graph of the height function of the passenger car for one turn of the wheel. Provide appropriate labels on the axes
c. Explain how you can identify the radius of the wheel from the graph in part (b).
d. If the center of the wheel is 55 feet above the ground, how high is the passenger car above the ground when it is at the top of the wheel?

Exercises 1–3

1. Each point 𝑃1, 𝑃2, … 𝑃8 on the circle in the diagram to the right represents a passenger car on a Ferris wheel. a. Draw segments that represent the co-height of each car. Which cars have a positive co-height? Which cars have a negative co-height? b. List the points in order of increasing co-height; that is, list the point with the smallest co-height first and the point with the largest co-height last.

1. Suppose that the radius of a Ferris wheel is 100 feet and the wheel rotates counterclockwise through one turn. Define a function that measures the co-height of a passenger car as a function of the degrees of rotation from the initial 3 o’clock position.
a. What is the domain of the co-height function?
b. What is the range of the co-height function?
c. How does changing the wheel’s radius affect the domain and range of the co-height function?
2. For a Ferris wheel of radius 100 feet going through one turn, how do the domain and range of the height function compare to the domain and range of the co-height function? Is this true for any Ferris wheel?

Exploratory Challenge: The Paper Plate Model, Revisited
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 and co-height of a Ferris wheel car at various amounts of rotation, measured with respect to the horizontal and vertical lines through the center of the wheel. 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 horizontal and vertical lines through the center of the wheel on the card stock behind the plate as shown. We will measure the height and co-height as the displacement from the horizontal and vertical lines through the center of the plate.
b. Using the physical model you created with your group, record your measurements in the table, and then graph each of the two sets of ordered pairs (rotation angle, height) and (rotation angle, co-height) on separate coordinate grids. Provide appropriate labels on the axes.

Closing

• Why do you think we named the new function the co-height?
• How are the graphs of these two functions alike? How are they different?
• What does a negative value of the height function tell us about the location of the passenger car at various positions around a Ferris wheel? What about a negative value of the co-height function?

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