Question

When Cheryls cousin starts riding the heart goes up before coming back down and hitting the pavement. the diameter of the wheel is 20 inches. Cheryls cousin rides at a steady pace and the wheel makes 60 revolutions per minute. find the equation that models the height of the heart in inches as a function of time (x) in minutes

Answer 1

To model the height of the heart as a function of time, we need to establish a few key parameters based on the information provided.

1. **Diameter of the wheel**: 20 inches. This means the radius \( r \) is:
\[
r = \frac{20}{2} = 10 \text{ inches}
\]

2. **Revolutions per minute**: The wheel makes 60 revolutions per minute. This helps us calculate how much height changes with time.

3. **Circumference of the wheel**: The distance covered in one complete revolution is the circumference of the wheel, which is calculated as:
\[
C = 2\pi r = 2\pi(10) = 20\pi \text{ inches}
\]

4. **Height Function**: As the wheel turns, the height of the heart will oscillate in a sinusoidal manner due to the circular motion.

We can model this height using a sine function:
\[
h(t) = a \sin(b(t - d)) + k
\]
Where:
- \( a \) is the amplitude
- \( b \) affects the period of the function
- \( k \) is the vertical shift
- \( d \) is the horizontal shift (phase shift)

### Parameters:

- **Amplitude \( a \)**: The amplitude is equal to the radius of the wheel (the maximum height above and below the center line):
\[
a = 10 \text{ inches}
\]

- **Vertical shift \( k \)**: The center of the wheel is at \( 10 \) inches; thus, we set:
\[
k = 10 \text{ inches}
\]

- **Angular frequency \( b \)**: The wheel makes 60 revolutions per minute. The period \( T \) is given by:
\[
T = \frac{1 \text{ minute}}{60 \text{ revolutions}} = \frac{1}{60} \text{ minutes per revolution}
\]
The frequency in radians per minute is:
\[
b = \frac{2\pi \text{ radians}}{T} = 2\pi \times 60 = 120\pi
\]

- **Phase shift \( d \)**: If we assume that at \( t=0 \), the heart is at its highest point, we can just use the sine function without a horizontal shift. So, \( d = 0 \).

### Final Height Function:

Combining all these parameters, the height of the heart \( h(t) \) in inches as a function of time \( t \) in minutes will be:
\[
h(t) = 10 \sin(120\pi t) + 10
\]

This equation models the height of the heart as a function of time while Cheryl's cousin rides the bike.