To solve this, we’ll go through each part systematically.

(a) Finding the Angular Speed in Radians per Minute

1. Angular speed (ω) is given by: $\omega = \text{revolutions per minute} \times \text{radians per revolution}$
2. The fan completes $70$ revolutions per minute, and each revolution is $2\pi$ radians. Therefore: $\omega = 70 \times 2\pi = 140\pi \text{ radians per minute}$

So, the angular speed is: $\omega = 140\pi \text{ radians per minute}$

(b) Finding the Linear Speed at the Tip of a Blade in Miles per Hour

1. Linear speed $v$ of a point on a rotating object is related to its angular speed by: $v = r \cdot \omega$ where $r$ is the radius of the rotation in inches, and $\omega$ is the angular speed in radians per minute.

2. Since the radius $r$ is 18 inches, we substitute: $v = 18 \times 140\pi = 2520\pi \text{ inches per minute}$

3. Now, we need to convert this to miles per hour.

   • Convert inches per minute to feet per minute by dividing by 12: $\frac{2520\pi}{12} = 210\pi \text{ feet per minute}$

   • Convert feet per minute to miles per hour by multiplying by $\frac{60}{5280}$ (since there are 5280 feet in a mile and 60 minutes in an hour): $v = 210\pi \times \frac{60}{5280}$

4. Simplify this expression: $v = \frac{210 \times 60 \times \pi}{5280} = \frac{12600\pi}{5280}$

5. Reducing this fraction: $v = \frac{105\pi}{44} \approx 7.5 \text{ miles per hour (rounded to the nearest whole number)}$

Final Answers

• (a) The angular speed of the fan is $140\pi$ radians per minute.
• (b) The linear speed of the tip of a blade is approximately $8$ miles per hour (to the nearest whole number).

Would you like further clarification on any step or additional details?

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1. What is the significance of angular speed versus linear speed in practical scenarios?
2. How does the radius of the fan affect both angular and linear speeds?
3. How would the linear speed change if the fan were rotating at a different rate, say 100 revolutions per minute?
4. Why do we use radians in angular speed calculations instead of degrees?
5. How might this conversion process apply to other rotating objects, like a car wheel?

Tip: Always check units carefully during conversion steps to avoid errors, especially with angular and linear speeds!