https://file.aiquickdraw.com/mathbot/file/1726980479464pjk1w2ig.jpg

Let's break down the problem:

1. **Given Information:**
   - Radius of the fan \( r = 14 \) inches.
   - The fan turns at a rate of 67 revolutions per minute (RPM).
   - Conversion factor: \( 1 \) mile = 5280 feet.

### (a) Angular speed in radians per minute
The formula to calculate angular speed \( \omega \) in radians per minute is:

\[
\omega = 2\pi \times \text{RPM}
\]

Given that the RPM is 67, we can substitute this value into the formula:

\[
\omega = 2\pi \times 67
\]

### (b) Linear speed at the tip of the blade in miles per hour
The linear speed \( v \) is related to the angular speed by:

\[
v = r \times \omega
\]

Where:
- \( r \) is the radius in feet (since 1 inch = \( \frac{1}{12} \) feet, \( 14 \) inches = \( \frac{14}{12} \) feet).
- \( \omega \) is the angular speed in radians per minute.

To convert linear speed from feet per minute to miles per hour, we use the following conversion:

\[
v_{\text{mph}} = v_{\text{fpm}} \times \frac{60}{5280}
\]

Let me calculate these values for you.The results are:

(a) The angular speed of the fan is approximately **421 radians per minute**.

(b) The linear speed at the tip of the blade is approximately **6 miles per hour**.

Would you like any more details or have any questions?

Here are 5 related questions for further practice:
1. How would the angular speed change if the fan were rotating at 100 RPM instead of 67 RPM?
2. If the radius of the fan was doubled, how would the linear speed change?
3. Convert the angular speed into degrees per minute.
4. What is the linear speed at the tip of the blade in feet per second?
5. How would the linear speed change if the fan turned at 120 revolutions per minute?

**Tip:** Angular speed is always in radians per unit time, and it’s directly proportional to the number of revolutions per minute.

A ceiling fan has 14-inch blades (so the radius of the circular fan is 14 inches). Suppose the fan turns at a rate of 67 revolutions per minute. (a) Find the angular speed of the fan in radians per minute. (b) Find the linear speed of the tip of a blade in miles per hour. (Use the fact that 1 mi = 5280 ft.)

Learn how to calculate the angular speed in radians per minute and the linear speed in miles per hour for a ceiling fan with 14-inch blades rotating at 67 RPM. This problem involves key concepts like angular speed, linear speed, and unit conversion. Suitable for high school students (Grades 9-12).

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