Jane has a bike with a 22-inch rear wheel. The wheel sprocket is 3 inches in diameter and the pedal sprocket is 4 inches in diameter (the sprockets hold the chain.) How fast in miles per hour is she going if she pedals at a rate of 100 revolutions per minute?

Rear-wheel diamater: 22

Wheel sprocket on rear-wheel: 3

Pedal sprocket: 4
AngularSpeed: 100revs/min

R1= radius of rear
W1=angular speed of rear

R2=radius of pedal sprocket
W2=Angular speed of pedal sprocket

(R1)(W1)=(R2)(W2)<-----(EQ1)
Plug in the corresponding values, but before that convert the 100revs into radiance per hour since the problem is asking of "how fast miles/hour"

The angular speed of the Rear-Sprocket and The rear wheel is Equal. The reason is because the sprocket is located on the wheel thus
Angular speed(sprocket)=Angular speed(wheel)

After you get the angular speed of the Rear sprocket on Equation one (EQ1) you can calculate the distance it covers by using the S=R(Theta)

where in the theta must be in radiance* and the radius u must use must be the radius of the wheel since we are looking for the distance of the wheel given a certain angle and radius of course.

pedal sprocket diameter* 4 inches

Be careful of the diameter, u need to divide it by 2 to get the radius.

To determine how fast Jane is going in miles per hour, we need to use a few formulas and conversion factors. Firstly, we need to calculate the distance covered per revolution. Then, we'll calculate the distance she covers in one minute based on her pedal rate and the diameter of the pedal sprocket. Finally, we'll convert the distance per minute to miles per hour.

1. Calculate the distance covered per revolution:
Since the diameter of the rear wheel is given, we can calculate the circumference of the wheel by using the formula:
Circumference = π * diameter

Let's plug in the values:
Circumference = π * 22 inches

2. Calculate the distance covered in one minute:
Since Jane pedals at a rate of 100 revolutions per minute, we need to multiply the distance covered per revolution by the number of revolutions:
Distance per Minute = Circumference * Number of Revolutions

3. Convert the distance covered per minute to miles per hour:
To convert from inches to miles, we need to divide by the number of inches in a mile (which is 63,360 inches). Then, to convert from minutes to hours, we divide by the number of minutes in an hour (which is 60 minutes).

Distance per Hour = (Distance per Minute / (inches in a mile)) * (minutes in an hour)

Let's perform the calculations:

1. Circumference = π * 22 inches
Circumference ≈ 3.1416 * 22 inches
Circumference ≈ 69.1152 inches

2. Distance per Minute = Circumference * 100 revolutions
Distance per Minute = 69.1152 inches * 100
Distance per Minute = 6911.52 inches

3. Distance per Hour = (Distance per Minute / 63360 inches) * (60 minutes)
Distance per Hour = (6911.52 inches / 63360 inches) * 60
Distance per Hour ≈ 0.6893 miles per hour

So, if Jane pedals at a rate of 100 revolutions per minute, she is going approximately 0.6893 miles per hour.