Bike and Sprocket Problem

[Chaim]

You are riding a bike along a level road. Assume each wheel is 28 inches in diameter, the rear sprocket has radius 3 inches and the front sprocket has radius r inches. Suppose you are pedaling the front sprocket at the rate of 1.5 ^rev/_sec and your forward speed is 11 mph on the bike. What is the radius of the front sprocket?

Answer: 4.4 inches

So here are the key information:
Wheel is 28 inches in diameter
Rear sprocket has raidus 3 inches
Front sprocket has raidus r inches
Pedaling front sprocket at the rate of 1.5 ^rev/_sec
Forward speed is 11 mph
Find raidus of front sprocket

I'm having trouble putting all this information in my head, can anyone help me through with the steps?

 

[pappus]

You are riding a bike along a level road. Assume each wheel is 28 inches in diameter, the rear sprocket has radius 3 inches and the front sprocket has radius r inches. Suppose you are pedaling the front sprocket at the rate of 1.5 ^rev/_sec and your forward speed is 11 mph on the bike. What is the radius of the front sprocket?

Answer: 4.4 inches

So here are the key information:
Wheel is 28 inches in diameter
Rear sprocket has raidus 3 inches
Front sprocket has raidus r inches
Pedaling front sprocket at the rate of 1.5 ^rev/_sec
Forward speed is 11 mph
Find raidus of front sprocket

I'm having trouble putting all this information in my head, can anyone help me through with the steps?

1. Determine the number of revolutions per second of the rear wheel which is the same as the number of revolutions per second of the rear cogwheel:

\(\displaystyle 11\ \frac{mi}h~\longrightarrow~193.6\ \frac{in}s\)

which will give \(\displaystyle 2.2\ \frac{rev}s\)

2. Since the same length of the chain passes the front and the rear cogwheel per second you can say

\(\displaystyle circumference_{front} \cdot rev_{front} = circumference_{rear} \cdot rev_{rear}\)

or in short:

\(\displaystyle radius_{front} \cdot rev_{front} = radius_{rear} \cdot rev_{rear}\)

\(\displaystyle x \cdot 1.5 = 3 \cdot 2.2~\implies~x = 4.4\)

3. Remark: I'm not very used to those archaic measures like inches, feet, miles, fathoms or yards. So please check my arithmetic.