Question 1185859: How many minutes after five o’clock will the hands of the clock be perpendicular for the second time? Found 2 solutions by Edwin McCravy, greenestamps:Answer by Edwin McCravy(20054) (Show Source):
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
I have seen students solve this kind of problem by many different methods similar to the one shown by the other tutor.
Here is the method I like....
The hour hand makes 1 revolution every 12 hours; the minute hand makes 12. Since both rotate at constant rates, any particular angle between the two hands will be formed 11 times every 12 hours. That means the interval between successive times that a particular angle is formed is 12/11 hours.
The second time after 5 o'clock that the hands form a 90 degree angle, the minute hand is 90 degrees ahead of the hour hand.
The minute hand is 90 degrees ahead of the hour hand at 9:00.
So the time between 5 and 6 o'clock when the minute hand will be 90 degrees ahead of the hour hand is 8 times 12/11 hours after 9:00, which is 8 times 1/11 hour after 5:00.
