In this scenario, we are analyzing the motion of the heart as it moves in a circular path around the bike wheel.
Determining Which Function to Use:
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Position of the Heart: As the heart travels around the wheel, its vertical position can be modeled as a periodic function since it moves up and down in a smooth oscillation.
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Starting Position: If we place the heart in the middle (the initial position), as the wheel starts to rotate, the heart moves upwards first. This behavior suggests that a sine function can be used, as the sine function starts at zero and rises to a positive maximum at π/2 (90 degrees). In contrast, the cosine function starts at a maximum value (when the angle is zero).
Given that the heart is at the middle of the bike wheel at the beginning of the ride and goes up from there, we can conclude that a sine function is more appropriate for modeling the height of the heart over time.
Writing the Function:
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Height Calculation: The height of the heart can be modeled as follows, where \( A \) is the amplitude (the maximum height from the center), \( T \) is the period of the motion, and \( t \) is the time in minutes.
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Amplitude: The radius of the bike wheel, given the circumference (12 inches), can be calculated as: \[ \text{Radius} = \frac{\text{Circumference}}{2 \pi} = \frac{12}{2 \pi} \approx 1.91 \text{ inches} \] Therefore, the amplitude \( A \) is the radius, which is approximately \( 1.91 \) inches.
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Period: The wheel makes 70 revolutions per minute. The angular frequency \( \omega \) in radians per minute is: \[ \omega = 70 \times 2\pi = 140\pi \text{ radians per minute} \] The period \( T \) (the time for one complete revolution) is given by: \[ T = \frac{2 \pi}{\omega} = \frac{2 \pi}{140\pi} = \frac{1}{70} \text{ minutes} \]
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Function Form: The general form of the sine function is: \[ h(t) = A \sin(\omega t) \] Substituting the values we have:
- Amplitude \( A \) = 1.91
- Angular frequency \( \omega \) = \( 140\pi \)
Thus, the function for the height of the heart in terms of time \( t \) (in minutes) is: \[ h(t) = 1.91 \sin(140\pi t) \]
This function describes the vertical position of the heart as the bike wheel turns, starting from the middle point and oscillating between about 0 inches (when at the bottom) and approximately 3.82 inches (when at the top), which is twice the amplitude because it moves from the center upwards to the maximum height and back down to the starting point.