To analyze the situation, let's first summarize the information given:
- The diameter of the tire is 12 inches, which gives a radius of \( r = \frac{12}{2} = 6 \) inches.
- The tire makes 70 revolutions per minute.
Choosing the function:
Given the setup, we can think of this situation in terms of circular motion. The charm is attached to the front tire, and its vertical position will oscillate as the tire rotates.
- At \( t = 0 \) seconds, the tire is at its midline height when the heart charm is at the highest point of its upward rotation.
A cosine function starts at its maximum value when \( t = 0 \), so it is a good fit for this scenario if we consider the vertical height of the charm relative to time.
Modeling the situation:
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Height of the charm: The vertical height \( h(t) \) of the charm can be modeled as a cosine function since it starts at its maximum height at \( t = 0 \).
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Amplitude: The amplitude corresponds to the radius of the tire, which is 6 inches.
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Vertical midline: The height of the tire relative to the ground will oscillate around the midline of its position. This position is 6 inches above the bottom of the tire, which can be considered at 0 inches — hence, we have a vertical shift.
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Angular frequency: The circumference of the tire can be found using the formula: \[ C = \pi \times d = \pi \times 12 \text{ inches} = 12\pi \text{ inches} \] In one minute, the tire makes 70 revolutions, so it travels: \[ 70 \text{ revolutions} \times 12\pi \text{ inches/revolution} = 840\pi \text{ inches} \]
To find the height function per second, we note that 70 revolutions per minute is equivalent to: \[ \text{Revolutions per second} = \frac{70}{60} = \frac{7}{6} \text{ revolutions per second} \] The angular frequency \( \omega \) in radians per second is: \[ \omega = 2\pi \times \text{Revolutions per second} = 2\pi \times \frac{7}{6} = \frac{7\pi}{3} \text{ radians per second} \]
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Final function: The height function \( h(t) \) can be written as: \[ h(t) = 6 \cos\left(\frac{7\pi}{3}t\right) + 6 \] Here, the \( +6 \) accounts for the vertical shift so that the function oscillates between 0 and 12 inches.
Final Expression:
Thus, the function that models the height of the charm over time is: \[ h(t) = 6 \cos\left(\frac{7\pi}{3} t\right) + 6 \]
This function describes the vertical position of the heart charm attached to the bike tire as it rotates.