Question 1203146
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Answers:<ol type="a"><li><font color=red>1229.384 revolutions per minute (rpm)</font></li><li><font color=red>7724.444 radians per minute</font></li></ol>Each result is approximate.


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Work Shown for part (a)


1 mile = 5280 ft
1 hour = 60 min


circumference = pi*diameter
C = pi*d
C = pi*1.8
C = 1.8pi
1 revolution = 1.8pi feet


{{{
matrix(1,2,79,"mph")
=
(matrix(1,2,79,"mi")/matrix(1,2,1,"hr"))
*
(matrix(1,2,5280,"ft")/matrix(1,2,1,"mi"))
*
(matrix(1,2,1,"hr")/matrix(1,2,60,"min"))
*
(matrix(1,2,1,"rev")/matrix(1,2,1.8pi,"ft"))
}}}


{{{
matrix(1,2,79,"mph")
=
(matrix(1,2,79,cross("mi"))/matrix(1,2,1,"hr"))
*
(matrix(1,2,5280,"ft")/matrix(1,2,1,cross("mi")))
*
(matrix(1,2,1,"hr")/matrix(1,2,60,"min"))
*
(matrix(1,2,1,"rev")/matrix(1,2,1.8pi,"ft"))
}}} "miles" units cancel


{{{
matrix(1,2,79,"mph")
=
(matrix(1,2,79,"")/matrix(1,2,1,cross("hr")))
*
(matrix(1,2,5280,"ft")/matrix(1,2,1,""))
*
(matrix(1,2,1,cross("hr"))/matrix(1,2,60,"min"))
*
(matrix(1,2,1,"rev")/matrix(1,2,1.8pi,"ft"))
}}} "hours" units cancel


{{{
matrix(1,2,79,"mph")
=
(matrix(1,2,79,"")/matrix(1,2,1,""))
*
(matrix(1,2,5280,cross("ft"))/matrix(1,2,1,""))
*
(matrix(1,2,1,"")/matrix(1,2,60,"min"))
*
(matrix(1,2,1,"rev")/matrix(1,2,1.8pi,cross("ft")))
}}} "feet" units cancel


{{{
matrix(1,2,79,"mph")
=
(matrix(1,2,79,"")/matrix(1,2,1,""))
*
(matrix(1,2,5280,"")/matrix(1,2,1,""))
*
(matrix(1,2,1,"")/matrix(1,2,60,"min"))
*
(matrix(1,2,1,"rev")/matrix(1,2,1.8pi,""))
}}} 


After those cancellations, the only thing left will be "rev" up top and "min" down below
That gives the angular speed unit "revolutions per minute". This abbreviates to RPM.


The next step is to use a calculator, or scratch paper, to compute the following
{{{
matrix(1,2,79,"mph")
=
(matrix(1,2,79,"")/matrix(1,2,1,""))
*
(matrix(1,2,5280,"")/matrix(1,2,1,""))
*
(matrix(1,2,1,"")/matrix(1,2,60,"min"))
*
(matrix(1,2,1,"rev")/matrix(1,2,1.8pi,""))
}}} 


{{{
matrix(1,2,79,"mph")
=
matrix(1,2,(79*5280*1*1)/(1*1*60*1.8pi),rev/min)
}}} 


{{{
matrix(1,2,79,"mph")
=
matrix(1,2,(417120)/(108pi),rev/min)
}}} 


{{{
matrix(1,2,79,"mph")
=
matrix(1,2,1229.38351597207,rev/min)
}}} This value is approximate. I used my calculator's stored version of pi to get the most accuracy possible.


{{{
matrix(1,2,79,"mph")
=
matrix(1,2,1229.384,rev/min)
}}} After rounding to 3 decimal places.
This equation is valid <b><u>only</u></b> when the wheel diameter is 1.8 feet.


The approximate result is <font color=red>roughly 1229.384 rpm</font>.


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Work Shown for part (b)


Refer to the previous part when we got to the step {{{(417120)/(108pi)}}}
I'll use that to avoid decimal rounding error.


{{{
matrix(1,2,(417120)/(108pi),"rpm")
=
(matrix(1,2,417120,"rev")/matrix(1,2,108pi,"min"))
*
(matrix(1,2,2pi,"radians")/matrix(1,2,1,"rev"))
}}}


{{{
matrix(1,2,(417120)/(108pi),"rpm")
=
(matrix(1,2,417120,cross("rev"))/matrix(1,2,108pi,"min"))
*
(matrix(1,2,2pi,"radians")/matrix(1,2,1,cross("rev")))
}}} "Rev" units cancel


{{{
matrix(1,2,(417120)/(108pi),"rpm")
=
(matrix(1,2,417120,"")/matrix(1,2,108*cross(pi),"min"))
*
(matrix(1,2,2*cross(pi),"radians")/matrix(1,2,1,""))
}}} The pi's cancel


{{{
matrix(1,2,(417120)/(108pi),"rpm")
=
(matrix(1,2,417120,"")/matrix(1,2,108,"min"))
*
(matrix(1,2,2,"radians")/matrix(1,2,1,""))
}}} 


{{{
matrix(1,2,(417120)/(108pi),"rpm")
=
matrix(1,2,(417120*2)/(108*1),"rads/min")
}}} 


{{{
matrix(1,2,(417120)/(108pi),"rpm")
=
matrix(1,2,(834240)/(108),"rads/min")
}}} 


{{{
matrix(1,2,(417120)/(108pi),"rpm")
=
matrix(1,2,7724.44444444444,"rads/min")
}}} This is approximate. The 4's go on forever.


{{{
matrix(1,2,(417120)/(108pi),"rpm")
=
matrix(1,2,7724.444,"rads/min")
}}} After rounding to 3 decimal places.



Summary 
79 mph = <font color=red>1229.384 rpm</font> 
79 mph = <font color=red>7724.444 radians per minute</font>
Each decimal value is approximate. 
This applies <b><u>only</u></b> when the wheel diameter is 1.8 feet.
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