I always think of this sort of stuff in terms of forcing the the result to have the units you want and/or the only units that can possibly make sense.
In the first case, you are given $\frac{rotations}{minutes}$ and $feet=\frac{feet}{1}$ and you want to end up with velocity, which has the general form $\frac{distance}{time}$. In this case, the pertinent units for distance and time are feet and minutes, respectively. The key, though, is the following: one rotation equals $2\pi$ times the radius (in feet). That is, $$\frac{rotations}{minutes}=2\pi\cdot radius\cdot\frac{feet}{minutes}.$$
But remember that in the end we just want an expression for velocity; that is, $\frac{feet}{minutes}=$ something. That's simple, though. Just divide both sides of the above equation by $2\pi\cdot radius$.
The second and third ones are very similar, but angular velocity has units $\frac{rotations}{minutes}$.
The last thing you asked is a specific instance of this sort of dimensional analysis, except that you have to convert everything to miles in the end. Alternatively, you can think of it as follows: if the velocity is 50 feet/second and the angular speed is 100 revolutions/second, that means in one second, the apparatus is making 100 revolutions - and furthermore, that these 100 revolutions = 50 feet. From this you get that one revolution is 1/2 a foot. That means that the circumference of the circle drawn out by the motion of the apparatus is 1/2 a foot, meaning that the radius of the circle is $\frac{1}{2}\cdot\frac{1}{2\pi}$ feet. To turn this into miles, just divide by 5280. Hope that helps.
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?
Best Answer
Hints:
Given a fixed speed, $v$ and radius, $r$, then:
$$v = \omega r, \omega = \frac{v}{r}, \omega = \frac{2 \pi}{time-of-1-revolution}$$
Be careful to watch for units and conversions!
Regards