Lesson 4 More Ferris Wheels Solidify Understanding

Learning Focus

Graph sine functions of the form $h (t) = a \sin (b t) + d$ .

How can I represent the vertical motion of a rider on a Ferris wheel graphically?

How does changing the speed, height, or radius of the Ferris wheel affect the graph and the function equation?

Open Up the Math: Launch, Explore, Discuss

Previously, you calculated the height of a rider on a Ferris wheel at different times $t$ , where $t$ represented the elapsed time after the rider passed the position farthest to the right of the Ferris wheel.

Recall the following facts for the Ferris wheel:

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

You also previously found several data points for the height of the rider at different times. Because of the symmetry of the positions on the wheel, you realized you didn’t need to calculate all of the heights, just a few were enough. Here are a couple of points you calculated:

At $t = 2 seconds$ the rider is at a vertical height of $45 feet$ .
At $t = 6 seconds$ the rider is at a vertical height of $54 feet$ .

1.

Based only on the information you previously found, as well as any additional insights you might have about riding on Ferris wheels, sketch a graph of the vertical height of a rider on this Ferris wheel as a function of the time elapsed since the rider passed the position farthest to the right of the Ferris wheel. (We can consider this position as the rider’s starting position at time $t = 0$ .)

2.

Write the equation for the graph you sketched.

3.

Of course, Ferris wheels do not all have this same radius, center height, or angular speed. Describe a different Ferris wheel by changing some of the facts listed above. For example, you can change the radius of the wheel, the height of the center, the angular speed, or the amount of time it takes to complete one revolution. You can even change the direction of rotation from counterclockwise to clockwise. If you want, you can change more than one fact. Just make sure your description seems reasonable for the motion of a Ferris wheel.

Description of my Ferris wheel:

4.

Sketch a graph of the height of a rider on your Ferris wheel as a function of the time elapsed since the rider passed the position farthest to the right of the Ferris wheel.

5.

Write the equation of the graph you sketched.

6.

We began this task by considering the graph of the height of a rider on a Ferris wheel with a radius of $25 feet$ and center $30 feet$ off the ground, which makes one revolution counterclockwise every $20 seconds$ . How would your graph change if:

The radius of the wheel is larger or smaller?

The height of the center of the wheel is greater or smaller?

The wheel rotates faster or slower?

7.

Given: $h (t) = D + A \sin (B t)$

How does the equation of the rider’s height change if:

The radius of the wheel is larger or smaller?

The height of the center of the wheel is greater or smaller?

The wheel rotates faster or slower?

Pause and Reflect

8.

Write the equation of the height of a rider on each of the following Ferris wheels $t seconds$ after the rider passes the farthest right position.

a.

The radius of the wheel is $30 feet$ , the center of the wheel is $45 feet$ above the ground, and the angular speed of the wheel is $15 degrees$ per second counterclockwise.

b.

The radius of the wheel is $50 feet$ , the center of the wheel is at ground level (you spend half of your time below ground), and the wheel makes one revolution clockwise every $15 seconds$ .

Ready for More?

1.

Create a description of a Ferris wheel that rotates clockwise, rather than counterclockwise. Describe how you will account for this clockwise rotation graphically and algebraically, then sketch a graph and write an equation for your description.

2.

Create a description of a Ferris wheel that rotates counterclockwise for a while, comes to an abrupt halt, and then rotates clockwise for the same interval of time as the counterclockwise motion. Describe how you will account for this motion graphically and algebraically, then sketch a graph and write an equation for your description.

Takeaways

Key features of the sine graph:

Midline:

Amplitude:

Period:

Quick-graph points:

Shape of graph:

In the function $h (t) = a \sin (b t) + d$

The parameter $a$ changes by . For example, doubling $a$ .
The parameter $b$ changes by . For example, doubling $b$ .
The parameter $b$ , which represents the angular speed of rotation, is related to the period by .
The parameter $d$ changes by .

We have made the following observation about the sine function: $\sin (- θ) = - \sin (θ)$

Vocabulary

Lesson Summary

In this lesson, we learned how to represent circular motion using a description, an equation, and a graph. We related the parameters $a$ , $b$ , and $d$ in the equation $f (t) = a \sin (b t) + d$ to the description of a Ferris wheel, and to the midline, amplitude, and period of a sine graph.

Retrieval

1.

Graph the circle, ${(x + 3)}^{2} + {(y - 2)}^{2} = 16$ , on the grid.

2.

The quadrants in a coordinate grid are always labeled I, II, III, and IV. Are they labeled in a clockwise or counterclockwise direction?