Use this worksheet to record your answers to the questions or

Question

problems for the Modeling Periodic Phenomena Portfolio. When you are finished,
save this worksheet with your answers and submit it for a portfolio grade.
Where indicated, draw figures on a blank sheet of paper or on a sheet of graph paper.
Be sure to label each sheet so that your teacher knows which answer goes with which
question. You can scan these figures and submit them as individual documents with
your portfolio worksheet, or you can take pictures of them and insert the pictures on
the portfolio worksheet.

Question 1
Cheryl’s cousin is about to embark on a 50-mile bike ride. Cheryl draws a little heart
on the front wheel of her cousin’s bike for good luck.
When Cheryl’s cousin starts riding, the heart goes up before coming back down and
hitting the pavement. The diameter of the wheel is [choose a number between 20 –
27.5] inches. Cheryl’s cousin rides at a steady pace and the wheel makes a
frequency of [choose a number between 60 – 80] revolutions per minute. Find the
equation that models the height of the heart in inches as a function of time (x) in
minutes.
A. Make a list of the given information. Choose two numbers as indicated in the
problem
Make sense of the problem. On graph paper, sketch a graph that shows the
height of the heart in inches over time in minutes after a few revolutions and
then calculate the frequency. Do not worry about scaling the
x-axis, refer back to the graph on page 3. Note that electronic graphing is
not always necessary nor successful.
C. Decide whether you will write an equation that involves sine or cosine. Explain
your decision.
D. Persevere in solving the problem. Use your sketch of the graph along with the
given information to determine the values of a, b, c, and d. Explain how you
determined each value.
E. Write the equation that models the scenario using the values of a, b, c, and d.
Question 2
A. Predict the height of the heart exactly 30 seconds into the ride. Use your equation from Question 1. Show your work.
B. Predict the height of the heart exactly 5 minutes into the ride. Use your equation from Question 1. Show your work.

Answer 1

Modeling Periodic Phenomena Portfolio Worksheet

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Question 1

A. Given Information:

• Diameter of the wheel: 24 inches (chosen for this example, any value between 20-27.5 is acceptable)
• Radius of the wheel (r): \( \frac{24}{2} = 12 \) inches
• Frequency of revolutions: 70 revolutions per minute (chosen for this example, any value between 60-80 is acceptable)

Height of the heart as it moves along with the wheel will vary as a function of time.

1. Circumference of the wheel \( C = \pi \times \text{diameter} = \pi \times 24 \approx 75.4 \) inches

2. Period of one revolution (T):
   \[ T = \frac{1 \text{ minute}}{70 \text{ revolutions}} = \frac{60 \text{ seconds}}{70} \approx 0.857 \text{ seconds} \]

3. Angular frequency (b):
   \[ b = \frac{2\pi \text{ radians}}{T} = \frac{2\pi}{0.857} \approx 7.33 \text{ radians/second} \]

B. Sketch of the Graph: (Here, you would sketch a sinusoidal wave that begins at the maximum height (12) at time t=0 and oscillates down to 0 and back.)

• The maximum height is 12 inches (when the heart is at the top of the wheel).
• The minimum height is 0 inches (when the heart hits the pavement).

(You can label the axes with time in seconds on the x-axis and height in inches on the y-axis.)

C. Decision for Sine or Cosine:

• I will use the cosine function because at \( t = 0 \), the height is at its maximum (12 inches).
• Therefore, the general form of the cosine function is more suited since it starts at the maximum.

D. Determining Values of a, b, c, and d:

• \( a = 12 \) (the amplitude; it varies from 0 to 12 inches, so the amplitude is \( 12 - 0 = 12 \)).
• \( b \approx 7.33 \) (angular frequency previously calculated).
• \( c = 0 \) (phase shift; since there is no horizontal shift in the cycle).
• \( d = 6 \) (vertical shift; midline height is at 6 inches since it oscillates between 0 and 12).

E. The equation that models the scenario: \[ h(x) = 12 \cos(7.33 x) + 6 \] where \( h(x) \) is the height of the heart in inches at time \( x \) in minutes.

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Question 2

A. Predict the height of the heart exactly 30 seconds into the ride:

1. Convert 30 seconds into minutes: \( 30 \text{ seconds} = 0.5 \text{ minutes} \).

2. Substitute \( x = 0.5 \) into the equation: \[ h(0.5) = 12 \cos(7.33 \times 0.5) + 6 \] \[ h(0.5) = 12 \cos(3.665) + 6 \]

   Calculate \( \cos(3.665) \): \[ \cos(3.665) \approx -0.138 (using a calculator) \] \[ h(0.5) = 12 \times (-0.138) + 6 \approx -1.656 + 6 = 4.344 \text{ inches} \]

B. Predict the height of the heart exactly 5 minutes into the ride:

1. Substitute \( x = 5 \): \[ h(5) = 12 \cos(7.33 \times 5) + 6 \] \[ h(5) = 12 \cos(36.65) \]

   Calculate \( \cos(36.65) \): \[ \cos(36.65) \approx 0.166 \] \[ h(5) = 12 \times 0.166 + 6 \approx 1.992 + 6 = 7.992 \text{ inches} \]

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Feel free to adjust the numbers based on your own selections within the provided ranges. Please make sure to sketch the appropriate graphs separately as directed!