Question 1185697: Find the two times between 11 and 12 o'clock when the hands of the clock are at right angles to each other.
Answer by greenestamps(13209)   (Show Source):
You can put this solution on YOUR website!
Here is the way I like to solve this kind of problem. There are many others; perhaps other tutors will respond with different solution methods.
The minute hand makes twelve revolutions every 12 hours; the hour hand makes one. So given any angle between the two hands, that angle is formed 11 times in every 12-hour period. So the interval between successive times that a particular angle is formed is 12/11 hours, which is 1 hour, 5 minutes, 27 3/11 seconds (1:05:27 3/11).
For this problem, the minute hand is 90 degrees "behind" the hour hand at 3:00, so the time between 11:00 and 12:00 that the minute hand is 90 degrees behind the hour hand is 3:00, minus 3 times (1:05:27 3/11).
3:00 - 3(1:05:27 3/11)
3:00 - 3:16:21 9/11
11:43:38 2/11
ANSWER 1 (to the nearest second) 11:43:38
And the minute hand is 90 degree "ahead of" the hour hand at 9:00, so the time between 11:00 and 12:00 that the minute hand is 90 degrees ahead of the hour hand is 9:00, plus 2 times (1:05:27 3/11).
9:00+2(1:05:27 3/11)
9:00+2:10:54 6/11)
11:10:54 6/11
ANSWER 2: (to the nearest second) 11:10:55
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