10. At what speed is a bicyclist traveling when his 27 -inch diameter tires are rotating at an angular speed of $$ 5 \pi $$ radians per second, to the nearest mile per hour?

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1. The tire has diameter $$27$$ inches, so its radius is  
   $$
   r = \frac{27}{2} = 13.5 \text{ inches}.
   $$

2. The tangential speed $$v$$ is related to the angular speed $$\omega$$ by  
   $$
   v = r \omega.
   $$  
   Substituting $$r = 13.5$$ inches and $$\omega = 5\pi$$ radians per second gives  
   $$
   v = 13.5 \times 5\pi = 67.5\pi \text{ inches per second}.
   $$

3. To convert inches per second to miles per hour:
   - There are $$63,\!360$$ inches in a mile.
   - There are $$3,\!600$$ seconds in an hour.

So,  
   $$
   \text{Speed in mph} = 67.5\pi \times \frac{3600}{63360}.
   $$

4. Simplify the conversion factor:
   $$
   \frac{3600}{63360} = \frac{3600 \div 120}{63360 \div 120} = \frac{30}{528} = \frac{30 \div 6}{528 \div 6} = \frac{5}{88}.
   $$

5. Now, the speed in mph is  
   $$
   v = 67.5\pi \times \frac{5}{88} = \frac{337.5\pi}{88} \text{ mph}.
   $$

6. Approximating numerically:
   $$
   337.5\pi \approx 337.5 \times 3.1416 \approx 1059.3,
   $$
   so
   $$
   v \approx \frac{1059.3}{88} \approx 12.05 \text{ mph}.
   $$

7. Rounding to the nearest mile per hour, the bicyclist is traveling approximately  
   $$
   12 \text{ mph}.
   $$
The bicyclist is traveling approximately 12 miles per hour.

At what speed is a bicyclist traveling when his 27 -inch diameter tires are rotating at an angular
speed of radians per second, to the nearest mile per hour?

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At what speed is a bicyclist traveling when his 27 -inch diameter tires are rotating at an angular speed of radians per second, to the nearest mile per hour?