Final answer:
a. The linear speed of the bicycle is 1170π inches per minute. b. The speed of the bike is 0.056 miles per hour.
Step-by-step explanation:
a. The linear speed of the bicycle can be found by multiplying the angular speed (in radians per minute) by the radius of the wheel. The angular speed can be calculated by converting the pedal speed (in revolutions per minute) to angular speed (in radians per minute). The pedal speed of 195 rpm can be converted to angular speed by multiplying it by 2π, since there are 2π radians in one revolution. So, the angular speed is 195 rpm * 2π = 390π radians per minute. Using the formula for linear speed (v = ωr), where v is the linear speed, ω is the angular speed, and r is the radius of the wheel, we can calculate the linear speed as follows: v = 390π * 3 inches (radius of the front sprocket) = 1170π inches per minute.
b. To convert the linear speed from inches per minute to miles per hour, we need to convert inches to miles and minutes to hours. There are 12 inches in a foot, 5280 feet in a mile, and 60 minutes in an hour. So, the conversion factor from inches per minute to miles per hour is 1 / (12 * 5280 * 60). By multiplying the linear speed in inches per minute by this conversion factor, we can find the speed in miles per hour: 1170π * (1 / (12 * 5280 * 60)) = 0.056 miles per hour.