Question

The radius of the tires of a car is 18 inches, and they are revolving at the rate of 651 revolutions per minute. How fast is the car traveling in miles per hour?

Answer 1

So the car is moving at 651 revolutions per minute, with wheels of a radius of 18inches

so, one revolution, is just one go-around a circle, and thus 2π, 651 revolutions is just 2π * 651, or 1302π, the wheels are moving at that "angular velocity"

now, what's the linear velocity, namely, the arc covered per minute

well   $\bf v=rw\qquad \begin{cases} v=\textit{linear velocity}\\ r=radius\\ w=\textit{angular velocity}\\ ----------\\ r=18in\\ w=1302\frac{\pi }{min} \end{cases}\implies v=18in\cdot \cfrac{1302\pi }{min} \\\\\\ v=\cfrac{23436\pi\ in}{min}$

now, how much is that in miles/hrs?  well
let's keep in mind that, there are 12inches in 1foot, and 5280ft in 1mile, whilst 60mins in 1hr

thus   $\bf \cfrac{23436\pi\ in}{min}\cdot \cfrac{ft}{12in}\cdot \cfrac{mi}{5280ft}\cdot \cfrac{60min}{hr}\implies \cfrac{23436\cdot \pi \cdot 60\ mi}{12\cdot 5280\ hr}$

notice, after all the units cancellations, you're only left with mi/hrs