Between 3:00 and 4:00, find the time for the long hand and the hour hand to become in a straight line in the opposite direction.
Because the long hand moves 360° for one hour (60 minutes), it moves 360°/60=6 degrees in one minute.
Because the hour hand moves one clockface = 30 degrees for one hour, it moves 30°/60=0.5 degrees in one minute.
When the long hand catches up with the hour hand by 90 degrees and goes ahead 180 degrees more, the long hand and an hour hand become in a straight line in the opposite direction.
That is, it may be determined the time it takes for the minute hand to move 270°(90 ° +180 ° = 270 °) more than the hour hand.
In one minute, the long hand proceeds 5.5°(6 ° -0.5 ° = 5.5 °) more than the hour hand.
Answer
3 o’clock 540/11 minutes
Solution
At 3:00 the angle of the long hand and the hour hand is 90 degrees(30° × 3 = 60°) and the long hand will move after the hour hand.
Because the long hand moves 360° for one hour (60 minutes), it moves 360°/60=6 degrees in one minute.
Because the hour hand moves one clockface = 30 degrees for one hour, it moves 30°/60=0.5 degrees in one minute.
When the long hand catches up with the hour hand by 90 degrees and goes ahead 180 degrees more, the long hand and an hour hand become in a straight line in the opposite direction.
That is, it may be determined the time it takes for the minute hand to move 270°(90 ° +180 ° = 270 °) more than the hour hand.
In one minute, the long hand proceeds 5.5°(6 ° -0.5 ° = 5.5 °) more than the hour hand.
It takes 540/11 minutes(270 ° / 5.5 ° = 540/11 minutes) advancing by 270 °.
