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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.
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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.

Answer 1

Question 1

A. Given Information

• Diameter of the wheel: 24 inches (chosen from the range 20 – 27.5)
• Radius of the wheel: \( r = \frac{24}{2} = 12 \) inches
• Frequency of revolutions: 70 revolutions per minute (chosen from the range 60 – 80)

B. Graph and Frequency Calculation

1. Sketch of the Graph (not drawn here):

   • Draw a sine or cosine wave to represent the height of the heart over time.
   • Since the heart is at the top of the wheel when it starts at the topmost point, the cosine function will be more suitable.
   • The graph will oscillate between the maximum and minimum heights:
     • Maximum height = radius + ground = 12 + 0 = 12 inches
     • Minimum height = 0 inches

2. Frequency Calculation:

   • The frequency \( f \) is given as 70 revolutions per minute.
   • To convert this into a radian frequency \( b \): \[ b = 2\pi f = 2\pi \left(\frac{70}{60}\right) = \frac{140\pi}{60} \approx 7.33 \text{ radians per minute} \]

C. Choice of Function

• Function Choice: We’ll use the cosine function.
• Explanation: Since the heart (initially at the top) corresponds to the maximum height of the graph, starting with a cosine function is appropriate (cosine starts at its maximum value).

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

1. Amplitude (a):

   • Amplitude is equal to the radius of the wheel: \( a = 12 \)

2. Angular Frequency (b):

   • As calculated earlier: \( b \approx 7.33 \) radians per minute

3. Phase Shift (c):

   • Since we start at the maximum height without shifting, \( c = 0 \)

4. Vertical Shift (d):

   • The wheel doesn't raise above the ground, so \( d = 0 \)

E. Equation for the Model

Putting all these values into the general form of a cosine function \( h(x) = a \cdot \cos(bx + c) + d \): \[ h(x) = 12 \cdot \cos(7.33x) + 0 \] Simplifying gives: \[ h(x) = 12 \cos(7.33x) \]

Summary:

• The equation that models the height of the heart as a function of time \( x \) in minutes is: \[ h(x) = 12 \cos(7.33x) \]