Question 1185859
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The minute hand's angular speed is 360 degrees per hour. or 6 degrees per minute
The hour hand's angular speed is 360 degrees per 12 hours, which is equal to
30 degrees per hour, or 0.5 degrees per minute.  The angle between the hands is
either decreasing or increasing at the rate of 6-0.5 = 5.5 degrees per minute.
At 5 o'clock the angle between the hands is 150 degrees. The angle between the
hands is decreasing at 5.5 degrees per minute.   The hands will be together when
the angle between them decreases the entire 150 degrees to 0. That will be
150/5.5 = 27 3/11 minutes. [Notice that on the hands' way to being together from
5 o'clock, they were perpendicular at one instant.]  Then after the hands are
together, the angle between the hands begins increasing at 5.5 degrees per
minute. They will be 90 degrees apart in 90/5.5 = 16 4/11 minutes.  That will be
the second time after five o'clock they were perpendicular.  So the answer is
 
27 3/11 + 16 4/11 = 43 7/ll minutes.  

Edwin</pre>