MM - Modeling with Periodic and Cyclical Functions Lesson
Modeling with Periodic and Cyclical Functions
What mathematics do you see in the gigantic Ferris wheel? Let's practice.
Select the buttons to name the mathematics in the Ferris Wheel.
The Unit Circle
On February 11, 2008, Singapore opened a new observation wheel called the Singapore Flyer. At the time of its opening, this giant Ferris wheel was the tallest in the world. The Singapore Flyer consists of an observation wheel with a diameter of 150 meters atop a boarding terminal, giving the structure an overall height of 165 meters. Twenty-eight air-conditioned capsules rotate on the outside of the wheel to provide unobstructed views of the city. The wheel rotates at a constant rate of 26 centimeters per second. This is slow enough that the wheel does not need to stop for loading and unloading unless there are special passenger needs.
Describe how they will calculate the capsule's height above the ground. Draw and label your diagram an accurate diagram of the wheel showing the dimensions given above. Use a compass or other tool to accurate draw the circle.
SOLVE THESE PROBLEMS:
- At Time 0, the capsule is in the boarding terminal. How high above the ground is the capsule at this moment?
Answer:
To find the height of the capsule at the board terminal, you subtract the height of the diameter on top of a boarding terminal from the overall height of the structure.
165 meters -150 meters = 15meters
- After one minute, how far has the capsule traveled around the circumference of the wheel?
Answer:
Since the wheel rotates at a constant rate of 26 cm/sec (given in problem) and there are 60 seconds in a minute:
Notice the right triangle you can draw within your diagram.
Once you have the hypotenuse and the angle of rotation, you can use the cosine function to determine the height of the capsule. Don't forget to account for the 15-meter boarding terminal as you think about the height of a capsule off the ground. If the angle of rotation is greater than 90 degrees, you will need to add it to the angle of rotation.
Fill in the table below showing the height of a single capsule changing as it rotates counterclockwise from the boarding terminal around the wheel. To do this, first calculate the circumference of the wheel.
- How many minutes does it take a capsule to make one complete revolution around the wheel? (round to the nearest minute) Explain the process.
Answer:
The wheel is traveling at 26 centimeters per second and the distance it must travel is 47, 124 cm.
- Before completing the table, explain how the angle values provided in the table are correct.
Answer:
If the wheel takes 30 minutes to complete one revolution, then 1/30 of the circumference is passed through each minute. Therefore, a single capsule has rotated 1/30 of the 360° of the circle or 12°.
- The first inscribed angle that models the situation after one minute is shown. Use additional diagrams and inscribed right triangles to determine more values of the total height as a given capsule continues to rotate through one complete revolution. Use trigonometry to calculate the corresponding values of a and complete the table, finding the height of the capsule at the various intervals of time. Complete the chart and then scroll down to check your answers.
Create a graph showing the height changing as a given capsule rotates through one complete revolution of the wheel. Show at least 10 well-spaced data points on your graph.
The following graphic illustrates how you go from a circle to a graph for the sine function.
Use technology to model the height of a capsule as it continues to rotate around the wheel. Show at least 90 minutes of rotation (3 rotations). Create a graph of the data. On your graph, label the period and amplitude of the curve. How do these values on your mathematical model relate to the physical context of the Singapore Flyer?
Periodic functions are functions that repeat over and over, or cycle on a specific period. They are also known as cyclical functions. Sine and Cosine Functions are the most common periodic functions.
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