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.
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Diameter of the wheel: 20 inches. This means the radius \( r \) is: \[ r = \frac{20}{2} = 10 \text{ inches} \]
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Revolutions per minute: The wheel makes 60 revolutions per minute. This helps us calculate how much height changes with time.
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Circumference of the wheel: The distance covered in one complete revolution is the circumference of the wheel, which is calculated as: \[ C = 2\pi r = 2\pi(10) = 20\pi \text{ inches} \]
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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:
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Amplitude \( a \): The amplitude is equal to the radius of the wheel (the maximum height above and below the center line): \[ a = 10 \text{ inches} \]
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Vertical shift \( k \): The center of the wheel is at \( 10 \) inches; thus, we set: \[ k = 10 \text{ inches} \]
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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: \[ b = \frac{2\pi \text{ radians}}{T} = 2\pi \times 60 = 120\pi \]
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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.