# Radius word problem help?

It's about the sprockets and chain of a bicycle. The pedal sprocket has a radius of 4 in., the wheel sprocket has a radius of 2 in., and the wheel a radius of 13 in. The cyclist pedals at 36 rpm.

I already figured out that the rotations per minute of the wheel sprocket is 72 rpm, but what is the speed of the bicycle, assuming that the wheel turns at the same rate as the wheel sprocket?

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## 2 Answers

If we assume the tire rolls without slipping, then each rotation of the wheel equates to a distance traversed equal to the circumference of the wheel. If you know the wheel makes $72$ rotations per minute and that the wheel has a radius of $13$ inches, can you figure out the total distance traversed in one minute?

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The wheel sprocket is connected to the wheel itself, so it also spins at 72rpm. The bike's speed is $2 \pi *13 * 72$ in/min.

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That can't be right. That would make the speed 5881.06, which it's not... – Brandt Oct 29 '12 at 0:12
@Brandt The answer seems right. What units is the answer for the speed given in? – EuYu Oct 29 '12 at 0:14
5881 in/min is 8.2 ft/sec., which is a very reasonable speed for a bike, even a bit slow. – Lubin Oct 29 '12 at 0:25
@EuYu it's mi/hr – Brandt Oct 29 '12 at 0:42
@Brandt $5881.06$ in/min translates to roughly $5.57$ mi/hr. Is that what you have? – EuYu Oct 29 '12 at 1:38