Question

the tires of a bicycle have radius 10.0 in. and are turning at the rate of 230 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?
□ mph (type an integer or decimal rounded to the nearest tenth as needed.)

Explanation:

Step1: Find the circumference of the tire

The formula for the circumference of a circle is $C = 2\pi r$. Given $r = 10.0$ in, so $C=2\pi(10)=20\pi$ in.

Step2: Find the distance traveled per minute

The tire makes 230 revolutions per minute. The distance $d$ traveled per minute is the number of revolutions times the circumference. So $d = 230\times20\pi$ in/min.
$d=4600\pi$ in/min.

Step3: Convert inches per minute to feet per minute

Since 1 foot = 12 inches, we divide the distance in inches per minute by 12. So the distance in feet per minute is $\frac{4600\pi}{12}=\frac{1150\pi}{3}$ ft/min.

Step4: Convert feet per minute to miles per hour

There are 5280 feet in a mile and 60 minutes in an hour.
The speed $v$ in miles per hour is $v=\frac{\frac{1150\pi}{3}\times60}{5280}$ mph.
First, $\frac{1150\pi}{3}\times60 = 23000\pi$ ft/hour.
Then $v=\frac{23000\pi}{5280}$ mph.
$v=\frac{575\pi}{132}$ mph $\approx 13.7$ mph.

Answer:

$13.7$