Question

the tires of a bicycle have radius 12.0 in. and are turning at the rate of 215 revolutions per min. see the figure. how fast is the bicycle traveling in miles per hour? (hint: 5280 ft = 1 mi)
given that the tires are on the ground, how fast is the bicycle traveling?

Explanation:

Step1: Calculate the circumference of the tire

The formula for the circumference of a circle is $C = 2\pi r$. Given $r = 12.0$ in, so $C=2\pi\times12.0 = 24\pi$ in.

Step2: Find the distance traveled per minute

The tire makes 215 revolutions per minute. The distance $d$ traveled per minute is the number of revolutions times the circumference. So $d = 215\times24\pi$ in/min.
$d=215\times24\pi=5160\pi$ in/min.

Step3: Convert inches - per - minute to feet - per - minute

Since 1 foot = 12 inches, the distance in feet per minute is $\frac{5160\pi}{12}= 430\pi$ ft/min.

Step4: Convert feet - per - minute to miles - per - hour

Since 1 mile = 5280 ft and 1 hour = 60 minutes.
The speed $v$ in miles per hour is $v=\frac{430\pi\times60}{5280}$ mi/h.
$v=\frac{25800\pi}{5280}=\frac{430\pi}{88}\approx\frac{430\times3.14}{88}=\frac{1350.2}{88}\approx15.34$ mi/h.

Answer:

Approximately $15.34$ mi/h