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The problem asks us to find the linear speed of a wheel with a diameter of 46 inches rotating at 19 degrees per second, and the result needs to be rounded to three decimal places.

### Solution:

The relationship between angular speed and linear speed is given by:
\[
v = r \cdot \omega
\]
Where:
- \(v\) is the linear speed,
- \(r\) is the radius of the wheel (half the diameter),
- \(\omega\) is the angular speed in radians per second.

### Steps:

1. **Convert the diameter to radius**:
   \[
   r = \frac{\text{diameter}}{2} = \frac{46}{2} = 23 \text{ inches}
   \]

2. **Convert the angular speed from degrees per second to radians per second**:
   Since \(360^\circ = 2\pi\) radians, we convert degrees to radians by multiplying by \(\frac{\pi}{180}\):
   \[
   \omega = 19^\circ/\text{s} \times \frac{\pi}{180} = \frac{19\pi}{180} \text{ radians/second}
   \]
   \[
   \omega \approx 0.3316 \text{ radians/second}
   \]

3. **Calculate the linear speed**:
   \[
   v = r \cdot \omega = 23 \cdot 0.3316 \approx 7.6268 \text{ inches/second}
   \]

4. **Round the result**:
   \[
   v \approx 7.627 \text{ inches/second}
   \]

### Final Answer:
The linear speed is approximately **7.627 inches/second**.

Would you like more details or have any questions? Here are some related questions to expand on the concept:

1. How does angular velocity relate to linear velocity?
2. What is the formula for converting between degrees and radians?
3. How would the linear speed change if the wheel’s diameter increased?
4. What effect does doubling the angular speed have on the linear speed?
5. How can the relationship between rotation and distance traveled be applied in real-world scenarios?

**Tip**: Remember that converting between degrees and radians is essential when working with circular motion problems involving angular and linear speeds!

A wheel with diameter 46 inches is rotating 19°/sec. Find the linear speed and round to 3 decimal places.

Find the linear speed of a wheel with a diameter of 46 inches that is rotating at 19°/sec. Using the relationship between angular and linear speed, this solution demonstrates how to convert angular speed from degrees to radians and calculate linear speed.

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