Lesson 5 Moving Shadows Practice Understanding

Jump Start

Swinging in her car on a Ferris wheel while waiting for the last car to load, Anahita was thinking about her friends in her math class. “I’m going to sketch a graph of my ride as soon as I get off, so I can remember to tell my friends all of the details of my ride.”

This is the graph Anahita made after exiting the ride.

a curved line is graphed on a coordinate plane with a beginning point of (0,70) x555101010151515202020252525y101010202020303030404040505050606060707070000

1.

Name one thing you notice about Anahita’s graph and provide an explanation of what that means.

2.

Name another thing you notice about Anahita’s graph and provide an explanation of what that means.

3.

Name a third thing you notice about Anahita’s graph and provide an explanation of what that means.

Learning Focus

Extend the definition of cosine to include all angles of rotation, and use cosine functions to model a context.

How can I describe the horizontal motion of the rider on the Ferris wheel both graphically and symbolically?

What are my steps for sketching sine and cosine graphs?

Open Up the Math: Launch, Explore, Discuss

In spite of his nervousness, Carlos enjoys his first ride on the amusement park Ferris wheel. He does, however, spend much of his time with his eyes fixed on the ground below him. After a while, he becomes fascinated with the fact that since the sun is directly overhead, his shadow moves back and forth across the ground beneath him as he rides around on the Ferris wheel.

Recall the following facts for the Ferris wheel Carlos is riding:

  • The Ferris wheel has a radius of .

  • The center of the Ferris wheel is above the ground.

  • The Ferris wheel makes one complete rotation counterclockwise every .

To describe the location of Carlos’s shadow as it moves back and forth on the ground beneath him, we could measure the shadow’s horizontal distance (in feet) to the right or left of the point directly beneath the center of the Ferris wheel, with locations to the right of the center having positive value and locations to the left of the center having negative values. For instance, in this system Carlos’s shadow’s location will have a value of when he is at the position farthest to the right on the Ferris wheel, and a value of when he is at a position farthest to the left.

1.

What would Carlos’s position be on the Ferris wheel when his shadow is located at in this new measurement system?

2.

Sketch a graph of the horizontal location of Carlos’s shadow as a function of time , where represents the elapsed time after Carlos passes position , the farthest right position on the Ferris wheel.

a blank 17 by 17 grid

3.

Calculate the location of Carlos’s shadow at the times, , given in the following table, where represents the number of seconds since Carlos passed the position farthest to the right on the Ferris wheel. Keep track of any regularities you notice in the ways you calculate the location of the shadow. As you calculate each location, plot Carlos’s position on the Ferris wheel diagram.

A circle within a circle representing a ferris wheel. The ferris wheel is divided into 10 equal parts with corresponding points labeled with the letters A through J. A right triangle is drawn between the center point, point a, and point b. The inner angle is 36 degrees, the hypotenuse is 25 feet, and the height is 25 times sine of 36 degrees. ABCDEFGHIJ36°25 ft25sin(36°)

Elapsed time since passing position

Calculations

Horizontal position of the rider

Pause and Reflect

4.

Write a general formula for finding the location of the shadow at any instant in time.

Quick Graphs

For each of the following, plot points on the midline to determine the period, and plot minimum and maximum points to find the range. Use these points to sketch two periods of the graph. Label the axes.

5.

a blank coordinate plane

Midline:

Maximum:

Minimum:

6.

a blank coordinate plane

Midline:

Maximum:

Minimum:

7.

a blank coordinate plane

Midline:

Maximum:

Minimum:

8.

a blank coordinate plane

Midline:

Maximum:

Minimum:

Ready for More?

Create a description of two Ferris wheels with the same radius, height of center, and angular speed, but one wheel has clockwise rotation, while the other has counterclockwise rotation.

1.

How will you account for the vertical motion under a clockwise rotation graphically and algebraically? Sketch the graph and write the equation for both of your Ferris wheels.

a.

Counterclockwise rotation

Graph:

a blank coordinate plane

Equation:

b.

Clockwise rotation

Graph:

a blank coordinate plane

Equation:

2.

How will you account for the horizontal motion under a clockwise rotation graphically and algebraically? Sketch the graph and write the equation for both of your Ferris wheels.

a.

Counterclockwise rotation

Graph:

a blank coordinate plane

Equation:

b.

Clockwise rotation

Graph:

a blank coordinate plane

Equation:

3.

Why does the sine graph get reflected over its midline, but the cosine graph does not?

Takeaways

To define cosine for an angle of rotation :

We can make observations about the sign of the sine and cosine in each of the quadrants of the circle, as seen in the diagram.

We have made the following observation about the cosine function:

A circle is graphed on a coordinate plane with a line that starts in the center and leaves the graph creating an obtuse angle in the circle initial rayterminal ray

Lesson Summary

In this lesson, we learned how to graph the horizontal position of a rider on the Ferris wheel using the cosine function. This required that we extend the definition of the cosine to include all angles of rotation. We examined attributes of the cosine graph and its equation and related those to similar attributes of the sine function.

Retrieval

1.

The wheel in the figure has evenly spaced spokes. Each spoke is .

a circle sliced into 8 even pieces with point C being in the center and a radius 12 inches

a.

Find the circumference.

b.

Find the distance between points and .

c.

Find .

d.

How many feet would the wheel roll if it made ?

2.

Perform the indicated operation.

a.

b.