If you are riding a bike and your front sprocket (the one with the pedals) has 40 teeth and the rear sprocket (the one connected to your back tire) has 20 teeth, and you are pedaling at a rate of 50 rpm, how fast are you going if your bike has 27 inch diameter wheels?

Assuming the teeth are equally spaced, the circumference of the front sprocket is twice that of the real sprocket, or

for each turn of the front we have two turns of the rear
Then at 50 rpm, the rear will will turn 100 rpm

circumf of rear wheel = 27π inches
distance covered in 1 minute (from rpm=rotations per minute)

27π(100) inches
or 2700π inches per minute
= 706.858.. ft/min
= appr 8.03 mph

To determine how fast you are going, we need to calculate the gear ratio and then use it to find the wheel's rotations per minute (RPM). From there, we can convert the RPM to speed in miles per hour (mph).

First, let's calculate the gear ratio by dividing the number of teeth on the front sprocket by the number of teeth on the rear sprocket:

Gear Ratio = Number of Teeth on Front Sprocket / Number of Teeth on Rear Sprocket

In this case, the gear ratio would be 40 / 20 = 2.

Next, let's calculate the wheel's RPM. We know you are pedaling at a rate of 50 rpm, but the rear sprocket is connected to the back tire. Since the front and rear sprockets are connected, they rotate at the same speed. Therefore, the wheel's RPM is also 50.

Now, to find the speed in mph, we need to convert the wheel's RPM to rotations per hour (RPH) and then multiply by the circumference of the wheel.

RPH = Wheel's RPM * 60

In this case, RPH = 50 * 60 = 3000.

To find the speed, we can use the following formula:

Speed (mph) = (RPH * Wheel Circumference) / 5280

The wheel circumference can be calculated using the diameter:

Wheel Circumference = π * Diameter

Let's substitute the values:

Wheel Circumference = π * 27 inches

Now, let's calculate the speed:

Speed (mph) = (3000 * π * 27) / 5280

Calculating this equation gives us the speed in mph.