Question 1203363
<font color=black size=3>
1 hour = 60 min
5280 feet = 1 mile
12 inches = 1 ft
1 revolution = 48pi  inches
2pi radians = 1 revolution


The 48pi refers to the circumference of the circle of diameter 48 inches.
C = pi*diameter


Each of those equivalences then form a fraction as shown in this long conversion string
{{{
(matrix(1,2,50,"mi")/matrix(1,2,1,"hr"))
(matrix(1,2,1,"hr")/matrix(1,2,60,"min"))
(matrix(1,2,5280,"ft")/matrix(1,2,1,"mi"))
(matrix(1,2,12,"in")/matrix(1,2,1,"ft"))
(matrix(1,2,1,"rev")/matrix(1,2,48pi,"in"))
(matrix(1,2,2pi,"rad")/matrix(1,2,1,"rev"))
}}}
The placement of each item is to form the next set of cancellations as I'll describe below.


{{{
(matrix(1,2,50,cross("mi"))/matrix(1,2,1,"hr"))
(matrix(1,2,1,"hr")/matrix(1,2,60,"min"))
(matrix(1,2,5280,"ft")/matrix(1,2,1,cross("mi")))
(matrix(1,2,12,"in")/matrix(1,2,1,"ft"))
(matrix(1,2,1,"rev")/matrix(1,2,48pi,"in"))
(matrix(1,2,2pi,"rad")/matrix(1,2,1,"rev"))
}}} "miles" units cancel


{{{
(matrix(1,2,50,"")/matrix(1,2,1,cross("hr")))
(matrix(1,2,1,cross("hr"))/matrix(1,2,60,"min"))
(matrix(1,2,5280,"ft")/matrix(1,2,1,""))
(matrix(1,2,12,"in")/matrix(1,2,1,"ft"))
(matrix(1,2,1,"rev")/matrix(1,2,48pi,"in"))
(matrix(1,2,2pi,"rad")/matrix(1,2,1,"rev"))
}}} "hours" units cancel


{{{
(matrix(1,2,50,"")/matrix(1,2,1,""))
(matrix(1,2,1,"")/matrix(1,2,60,"min"))
(matrix(1,2,5280,cross("ft"))/matrix(1,2,1,""))
(matrix(1,2,12,"in")/matrix(1,2,1,cross("ft")))
(matrix(1,2,1,"rev")/matrix(1,2,48pi,"in"))
(matrix(1,2,2pi,"rad")/matrix(1,2,1,"rev"))
}}} "feet" units cancel


{{{
(matrix(1,2,50,"")/matrix(1,2,1,""))
(matrix(1,2,1,"")/matrix(1,2,60,"min"))
(matrix(1,2,5280,"")/matrix(1,2,1,""))
(matrix(1,2,12,cross("in"))/matrix(1,2,1,""))
(matrix(1,2,1,"rev")/matrix(1,2,48pi,cross("in")))
(matrix(1,2,2pi,"rad")/matrix(1,2,1,"rev"))
}}} "inches" units cancel


{{{
(matrix(1,2,50,"")/matrix(1,2,1,""))
(matrix(1,2,1,"")/matrix(1,2,60,"min"))
(matrix(1,2,5280,"")/matrix(1,2,1,""))
(matrix(1,2,12,"")/matrix(1,2,1,""))
(matrix(1,2,1,cross("rev"))/matrix(1,2,48pi,""))
(matrix(1,2,2pi,"rad")/matrix(1,2,1,cross("rev")))
}}} "revolution" units cancel


There's a lot of cancellations going on. 
After the dust settles, we have "radians" up top and "minutes" in the bottom.
Those are the only two units left.


Multiply the values along the top: 50*1*5280*12*2pi = 6,336,000pi


Multiply the values along the bottom: 1*60*1*1*48pi*1 = 2880pi


We end up with {{{(6336000pi)/(2880pi) = 2200}}}


Answer to part (a) is <font color=red>2200 radians per minute</font>


=======================================================


Part (b)


We'll have this conversion string
{{{
(matrix(1,2,50,"mi")/matrix(1,2,1,"hr"))
(matrix(1,2,1,"hr")/matrix(1,2,60,"min"))
(matrix(1,2,5280,"ft")/matrix(1,2,1,"mi"))
(matrix(1,2,12,"in")/matrix(1,2,1,"ft"))
(matrix(1,2,1,"rev")/matrix(1,2,48pi,"in"))
}}}
The key difference is we don't involve "2pi radians = 1 rev" at the very end. 


So we haven't multiplied by 2pi. We can divide the previous result 2200 rad/min by 2pi to get the rpm (revolutions per min)


rpm = (rads per min)/(2pi)
rpm = (2200)/(2pi)
<font color=red>rpm = 350.14087480217 approximately</font>


I used the calculator's stored version of pi to get the most accuracy possible.
If your teacher instructs you to use something like pi = 3.14, then be sure to do so. Round the final answer however your teacher instructs.
</font>