A rider in a road bike race had an average linear speed of 39.532 ​km/hr. The wheel of his bicycle had a diameter of 68 cm. What is his angular​ speed, in revolutions per​ hour?(Do not round until the final answer. Then round to the nearest integer as​ needed.)

To find the rider's angular speed in revolutions per hour, we can follow these steps:

### Step 1: Relate linear speed and angular speed
The relationship between linear speed \((v)\) and angular speed \((\omega)\) is given by the formula:

\[
v = \omega \times r
\]
where:
- \(v\) is the linear speed (in km/hr),
- \(\omega\) is the angular speed in radians per hour,
- \(r\) is the radius of the wheel in kilometers.

### Step 2: Calculate the radius of the wheel
The diameter of the wheel is given as 68 cm. The radius is half of that:

\[
r = \frac{68 \, \text{cm}}{2} = 34 \, \text{cm}
\]

Convert this into kilometers:

\[
r = \frac{34}{100000} = 0.00034 \, \text{km}
\]

### Step 3: Solve for angular speed in radians per hour
Now, using the equation \(v = \omega \times r\):

\[
39.532 = \omega \times 0.00034
\]

Solving for \(\omega\):

\[
\omega = \frac{39.532}{0.00034} \approx 116273.53 \, \text{radians per hour}
\]

### Step 4: Convert angular speed to revolutions per hour
Since 1 revolution is \(2\pi\) radians, we can convert radians per hour to revolutions per hour by dividing by \(2\pi\):

\[
\text{Revolutions per hour} = \frac{116273.53}{2\pi} \approx \frac{116273.53}{6.28318} \approx 18507 \, \text{revolutions per hour}
\]

### Final Answer:
The rider's angular speed is approximately **18,507 revolutions per hour**.

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Would you like more details or have any questions on this problem?

Here are 5 related questions to explore:
1. How would the angular speed change if the diameter of the wheel increased?
2. What is the angular speed in revolutions per second?
3. If the linear speed of the rider decreased to 30 km/h, what would be the new angular speed?
4. How can we convert revolutions per hour to radians per second?
5. How many total revolutions does the wheel make in 2 hours at this speed?

**Tip:** When converting between linear and angular motion, always ensure the units are consistent—especially for radius and speed!

A rider in a road bike race had an average linear speed of 39.532 ​km/hr. The wheel of his bicycle had a diameter of 68 cm. What is his angular​ speed, in revolutions per​ hour? (Do not round until the final answer. Then round to the nearest integer as​ needed.)

Discover how to calculate the angular speed of a bicycle wheel with a linear speed of 39.532 km/hr and a diameter of 68 cm. This detailed solution uses key concepts in angular and linear speed, unit conversions, and the relationship between radians and revolutions. Suitable for high school students in Grades 10-12.

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