To determine whether a sine or cosine function is more appropriate, we need to think about the position of the charm on the bike wheel in relation to time.
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Understanding the Wheel's Motion:
- As the bike wheel rotates, the charm will rise up and down in a circular motion, making a complete cycle with each revolution of the wheel.
- If we assume that at \( t = 0 \) (the beginning of the time interval) the charm is at its highest point, the cosine function, which starts at its maximum value, would be suitable. Conversely, the sine function starts at \( 0 \).
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Setting up the parameters:
- The radius of the wheel is half the diameter:
\[ \text{Radius} = \frac{24 \text{ in}}{2} = 12 \text{ in} \] - The wheel makes 70 revolutions per minute, so we can calculate the angular frequency \( \omega \): \[ \text{Revolutions per minute} = 70 \text{ rev/min} \] Since one full revolution corresponds to \( 2\pi \) radians, the angular frequency in radians per minute is: \[ \omega = 70 \cdot 2\pi \text{ radians/min} = 140\pi \text{ radians/min} \]
- The radius of the wheel is half the diameter:
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Writing the Function:
- The vertical position \( y(t) \) of the charm above the ground (where \( y=0 \) is the ground level) can be modeled as: \[ y(t) = R \cdot \cos(\omega t) + d \]
- Here, \( R \) is the radius (12 inches), \( \omega \) is the angular frequency (\( 140\pi \) radians/min), and \( d \) is the vertical shift, which would be the radius itself if we want the charm to oscillate around the ground level. Since the lowest point of the charm when it hits the ground is at \( 0 \), we need to add the radius: \[ d = 12 \]
- Thus, the function becomes: \[ y(t) = 12 \cdot \cos(140\pi t) + 12 \]
Therefore, the cosine function is indeed a better fit for modeling the motion of the charm, leading to the final model: \[ y(t) = 12 \cdot \cos(140\pi t) + 12 \]
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