# Lesson 18

Modeling Circular Motion

- Let's use trigonometric functions to model circular motion.

### Problem 1

Jada is riding on a Ferris wheel. Her height, in feet, is modeled by the function \(h(m) = 100\sin\left(\text-\frac{\pi}{2} + \frac{2\pi m}{10}\right) + 110\), where \(m\) is the number of minutes since she got on the ride.

- How many minutes does it take the Ferris wheel to make one full revolution? Explain how you know.
- What is the radius of the Ferris wheel? Explain how you know.
- Sketch a graph of \(h\).

### Problem 2

The vertical position, in feet, of the point \(P\) on a windmill is represented by \(y = 5\sin\left(\frac{2\pi t}{3}\right)+20\), where \(t\) is the number of seconds after the windmill started turning at a constant speed. Select **all** the true statements.

The windmill blades are 5 feet long.

The windmill blades make 5 revolutions per second.

The midline for the graph of the equation is 20.

The windmill makes one revolution every 3 seconds.

The windmill makes 3 revolutions per second.

### Problem 3

A seat on a Ferris wheel travels \(250\pi\) feet in one full revolution. How many feet is the carriage from the center of the Ferris wheel?

\(\frac{125}{\pi}\)

\(\frac{250}{\pi}\)

125

250

### Problem 4

A carousel has a radius of 20 feet. The carousel makes 8 complete revolutions.

- How many feet does a person on the carousel travel during these 8 revolutions?
- What angle does the carousel travel through?
- What is the relationship between the angle of rotation and the distance traveled on this carousel? Explain your reasoning.

### Problem 5

- For which angle measures between 0 and \(2\pi\) is the cosine negative and the sine positive?
- For which angle measures between 0 and \(2\pi\) is the cosine negative and the sine negative?

### Problem 6

A \(\frac{\pi}{2}\) radian rotation takes a point \(D\) on the unit circle to a point \(E\). Which other radian rotation also takes point \(D\) to point \(E\)?

\(\frac{3\pi}{2}\)

\(\frac{4\pi}{2}\)

\(\frac{5\pi}{2}\)

\(\frac{7\pi}{2}\)

### Problem 7

A windmill blade spins in a counterclockwise direction, making one full revolution every 5 seconds.

Which statements are true? Select **all** that apply.

After 15 seconds, the point \(W\) will be in its starting position.

After \(\frac{1}{5}\) of a second, the point \(W\) will be in its starting position.

In 1 second, the point \(W\) travels through an angle of \(\frac{\pi}{5}\).

The position of \(W\) repeats every 5 seconds.

The position of \(W\) repeats every 10 seconds.

### Problem 8

Here is the graph of a trigonometric function.

Which equation has this graph?

\(y = \text-2\sin(2x)\)

\(y = 2\sin\left( 2\pi \left(x+\frac{1}{4}\right)\right)\)

\(y = 2\sin\left( 2\pi \left(x-\frac{1}{4}\right)\right)\)

\(y = 2\sin\left( 2\pi \left(x-\frac{\pi}{4}\right)\right)\)