We are given the following information:

• The initial angular speed of the bike's wheel is $\omega_0 = \frac{1 \, \text{revolution}}{6 \, \text{seconds}}$.
• The final angular speed of the bike's wheel is $\omega_f = 18 \, \text{revolutions per minute}$.
• The time interval during which this change occurs is $t = 22.7 \, \text{seconds}$.

We want to determine the angle $\theta$ (in revolutions) that the wheel turns over this period.

Step 1: Convert all angular speeds to the same units.

• The initial angular speed is already in revolutions per second: $\omega_0 = \frac{1}{6} \, \text{revolutions per second}.$
• The final angular speed is given in revolutions per minute, so we convert it to revolutions per second: $\omega_f = \frac{18 \, \text{revolutions}}{60 \, \text{seconds}} = 0.3 \, \text{revolutions per second}.$

Step 2: Calculate the average angular speed.

The average angular speed over the time interval can be calculated as: $\omega_{\text{avg}} = \frac{\omega_0 + \omega_f}{2} = \frac{\frac{1}{6} + 0.3}{2} = \frac{0.1667 + 0.3}{2} = 0.23335 \, \text{revolutions per second}.$

Step 3: Calculate the angle turned.

The angle $\theta$ turned by the wheel during the time interval $t = 22.7 \, \text{seconds}$ is given by: $\theta = \omega_{\text{avg}} \cdot t = 0.23335 \times 22.7 = 5.299 \, \text{revolutions}.$

Thus, the wheel turns through an angle of approximately 5.30 revolutions over the 22.7 seconds.

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Would you like more details on the calculation, or do you have any specific questions?

Here are 5 related questions to deepen understanding:

1. How would the result change if the time interval was doubled?
2. What if the final angular speed was 25 revolutions per minute instead?
3. How does the average angular speed relate to the angular acceleration?
4. Can you find the total number of radians the wheel turns instead of revolutions?
5. How does constant angular acceleration affect the angle turned?

Tip: When converting between units like minutes and seconds, always ensure that both angular speeds are expressed in the same units before calculating.