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?

To find the speed of the bicyclist, we first need to determine the linear speed associated with the tire's angular speed. The formula for linear speed \( v \) is given by: \[ v = r \cdot \omega \] where \( r \) is the radius of the tire and \( \omega \) is the angular speed in radians per second. 1. Given that the diameter of the tire is 27 inches, the radius \( r \) is: \[ r = \frac{27}{2} = 13.5 \text{ inches} \] 2. The angular speed \( \omega \) is given as \( 5\pi \) radians per second. 3. We can now substitute these values into the linear speed formula: \[ v = 13.5 \cdot 5\pi \] 4. To find \( v \) in inches per second: \[ v = 67.5\pi \text{ inches per second} \approx 212.06 \text{ inches per second} \] 5. To convert inches per second to miles per hour, we use the conversion factors \( 1 \text{ mile} = 63,360 \text{ inches} \) and \( 1 \text{ hour} = 3600 \text{ seconds} \): \[ v \text{ (in mph)} = \left( \frac{67.5\pi}{63360} \right) \times 3600 \] 6. Calculating the speed in mph: \[ v \text{ (in mph)} \approx \left( \frac{67.5 \times 3.14}{63360} \right) \times 3600 \approx \left( \frac{211.8}{63360} \right) \times 3600 \approx 11.98 \text{ mph} \] Rounding to the nearest mile per hour, the bicyclist is traveling at: **12 mph.**