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.

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 $25 feet$ .
The center of the Ferris wheel is $30 feet$ above the ground.
The Ferris wheel makes one complete rotation counterclockwise every $20 seconds$ .

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 $25$ when he is at the position farthest to the right on the Ferris wheel, and a value of $- 25$ 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 $0$ in this new measurement system?

2.

Sketch a graph of the horizontal location of Carlos’s shadow as a function of time $t$ , where $t$ represents the elapsed time after Carlos passes position $A$ , the farthest right position on the Ferris wheel.

3.

Calculate the location of Carlos’s shadow at the times, $t$ , given in the following table, where $t$ 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.

Elapsed time since passing position $A$	Calculations	Horizontal position of the rider
$1 second$
$2 seconds$
$3 seconds$
$4 seconds$
$8 seconds$
$14 seconds$
$18 seconds$
$23 seconds$
$28 seconds$
$40 seconds$

Pause and Reflect

1.

The wheel in the figure has $8$ evenly spaced spokes. Each spoke is $12 inches$ .

a.

Find the circumference.

b.

Find the distance between points $A$ and $B$ .

c.

Find $m ∠ B C A$ .

d.

How many feet would the wheel roll if it made $16 revolutions$ ?

2.

Perform the indicated operation.

a.

$\frac{x^{2} - 36}{x^{2} - 10 x + 25} \cdot \frac{6 x - 30}{2 x + 12}$

b.

$\frac{5 x}{(x + 2)} + \frac{2 x}{(x + 3)}$