Question

The diameter of each wheel of a bicycle is 27 inches. If you are traveling at a speed of 20 miles per hour on this bicycle, through how many revolutions per minute are the wheels turning?

Answer 1

Notice, you've got a diameter in inches
and a linear velocity in miles
thus, you'd need to convert first

$\bf v=rw\implies \cfrac{v}{r}=w\qquad \begin{cases} v=\textit{linear speed}\\ r=radius\\ w=\textit{angular speed}\\ --------------\\ v=20\frac{mi}{h}\\ r=\frac{diameter}{2}=\frac{27}{2} \end{cases} \\\\ \textit{now, converting 20miles to inches} \\\\ \cfrac{20mi}{h}\cdot \cfrac{5280ft}{mi}\cdot \cfrac{12in}{ft}\implies \cfrac{20\cdot 5280\cdot 12\ in}{h}\impliedby v \\\\\\ \cfrac{1267200\frac{in}{h}}{\frac{27}{2}in}=w$

now, whatever that is, you'll need to divide it by a revolution, recall, a revolution is $2\pi$
thus $\bf \cfrac{\qquad \boxed{\cfrac{1267200\frac{in}{h}}{\frac{27}{2}in}}\qquad }{2\pi }=revolutions$

Answer 2

Data:
Speed = 20 miles/hour $\stackrel{converting}{\rightarrow}$ 352 inches/sec
diameter of wheel (length) = 27 inches

Solving:
Dividing the initial speed of 352 inches per second by the length of 27 inches we have:
$\frac{352\diagup\!\!\!\!\!in/sec}{27\diagup\!\!\!\!\!in} \approx 13.037\:revolutions\:per\:seconds$

Turning revolutions per second into revolutions per minute, we have:


revolutions per second_____revolutions per minute
1 $\stackrel{converting}{\longrightarrow}$ 60
13.037 $\stackrel{converting}{\longrightarrow}$ y

$\frac{1}{13.037} = \frac{60}{y}$
multiply cross
$1*y = 13.037*60$
$y = 782.22\to\:\boxed{\boxed{y\approx782\:revolutions\:per\:minute}}\end{array}}\qquad\quad\checkmark$