A bicycle with 24-inch diameter wheels is traveling at 15 mi/h. Find the angular speed of the wheels in rad/min. How many revolutions per minute do the wheels make?

To solve this, we will break down the problem into two parts: finding the angular speed in radians per minute and determining the number of revolutions per minute (RPM).

### Part 1: Angular Speed in Radians per Minute

1. **Convert the speed of the bicycle from miles per hour (mi/h) to inches per minute (in/min)**:
   - \( 1 \text{ mile} = 5280 \text{ feet} \)
   - \( 1 \text{ foot} = 12 \text{ inches} \)
   - Therefore, \( 1 \text{ mile} = 5280 \times 12 = 63,360 \text{ inches} \)
   - The bicycle is traveling at \( 15 \text{ mi/h} \), so convert this to inches per minute:
     \[
     15 \text{ mi/h} = 15 \times 63,360 = 950,400 \text{ inches per hour}
     \]
     \[
     \text{To get inches per minute: } \frac{950,400}{60} = 15,840 \text{ in/min}
     \]

2. **Calculate the circumference of the wheel**:
   - The diameter of the wheel is 24 inches, so the circumference \( C \) of the wheel is:
     \[
     C = \pi \times d = \pi \times 24 \approx 75.398 \text{ inches}
     \]

3. **Determine the number of wheel rotations per minute**:
   - To find how many wheel rotations happen per minute, divide the speed of the bicycle in inches per minute by the circumference of the wheel:
     \[
     \text{Revolutions per minute (RPM)} = \frac{15,840}{75.398} \approx 210 \text{ revolutions per minute}
     \]

4. **Convert RPM to angular speed in radians per minute**:
   - One revolution corresponds to \( 2\pi \) radians. Therefore, the angular speed in radians per minute is:
     \[
     \text{Angular speed} = 210 \times 2\pi \approx 1320 \text{ radians per minute}
     \]

### Final Answers:
- **Angular speed of the wheels**: approximately \( 1320 \text{ rad/min} \)
- **Revolutions per minute**: \( 210 \text{ RPM} \)

Would you like further details or have any questions?

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Here are 5 related questions to expand on this problem:
1. How would the angular speed change if the bicycle's speed increased to 20 mi/h?
2. What is the angular speed in radians per second instead of per minute?
3. How would the angular speed differ if the wheels had a 26-inch diameter instead of 24 inches?
4. How long does it take for the wheel to complete one full revolution?
5. What is the distance traveled by the bicycle in 10 minutes?

**Tip**: Remember, angular speed and linear speed are directly related through the wheel's radius or diameter. Smaller wheels will spin faster for the same linear speed!

Calculate the angular speed of a bicycle wheel with a 24-inch diameter traveling at 15 mi/h. The solution involves converting the bicycle's speed into inches per minute, determining the wheel's RPM, and calculating angular velocity in rad/min. Suitable for high school students in Grades 9-12.

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