Question 1208853: Tammy starts her aerobics exercises every day after 6am, when the hands of the clock first angle make a 90 degree angle. She finishes after 7am, when the hands of the clock make a 120 degree angle. How much time does Tammy spend exercising every week?
Found 2 solutions by greenestamps, KMST:
Answer by greenestamps(13209)   (Show Source):
You can put this solution on YOUR website!
Solution method #1....
Let x be the number of degrees the hour hand moves during her workout.
Draw a rough sketch showing the hands of the clock at both times on the same diagram to see that the number of degrees the minute hand moves is
(360-(120-90)+x) = 330+x
The minute hand moves 12 times as fast as the hour hand (the hour hand makes one revolution in 12 hours; the minute hand makes one revolution in 1 hour). So
330+x = 12x
330 = 11x
x = 30
The hour hand moves 30 degrees in 1 hour, so the duration of Tammy's exercises is 1 hour.
ANSWER: Tammy exercises for 1 hour
Solution method #2....
The minute hand makes 12 revolutions in half a day; the hour hand makes one revolution. Because of that, the times when the two hands make a particular angle are separated by intervals of 12/11 hours.
When she starts her workout shortly after 6am, the minute hand is 90 degrees behind the hour hand. The minute hand is also 90 degrees behind the hour hand at 3:00, so the time shortly after 6am when the minute hand is 90 degrees behind the hour hand is 3:00, plus 3 times 12/11 hours, which gives us a "time" of 6 3/11 hours.
When she finishes her workout shortly after 7am, the minute hand is 120 degrees behind the hour hand. The minute hand is also 120 degrees behind the hour hand at 4:00, so the time shortly after 7am when the minute hand is 120 degrees behind the hour hand is 4:00, plus 3 times 12/11 hours, which gives us a "time" of 7 3/11 hours.
So the time between the beginning and end of her workout is exactly 1 hour.
ANSWER: Tammy exercises for 1 hour
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(Added later....)
Having shown solutions by two different methods that I was familiar with in my initial response and finding the answer was 1 hour, I now see that there is a quite simple method for solving the problem.
In any 1-hour interval, the angle between the hands changes by 30 degrees, because the hour hand moves 30 degrees while the minute hand moves a full 360 degrees.
In this problem, since the angle between the hands changed by 30 degrees (from 90 to 120) during her workout, her workout must have been 1 hour.
ANSWER (again) 1 hour
Answer by KMST(5328)  (Show Source):
You can put this solution on YOUR website! During Tammy's exercise, the minute hand went from a position  (or 15 minutes) behind the hour hand
to a position  (or 20 minutes) behind the hour hand, lapping it once. That is a gain of  or  .
As the minute hand is 12 times faster, it covers 12 minutes around the clock in 12 minutes, while the hour hand travels a distance/angle of 1 minute.
The minute hand gains over the hour hand a distance of 11minutes every 12minutes.
That gain requires  = 
You can calculate based on degrees, but I find it easier to use minutes on the analog clock to measure the angles swept by the hour and minute hands.
THE STORY AS I SEE IT:
The minute hand and the hour hand run around the clock, like high school students running around the track, the faster ones lapping the slower ones.
When Tammy starts exercising it is a bit past 6:15. The minute hand is (15 minutes) behind the hour hand.
As the minute hand runs faster than the hour hand, that angle will becomes smaller and smaller until the minute hand catches up with the hour hand. Then the angle is  .
After that, the minute hand will be ahead of the hour hand, and the angle will start increasing.
By 7AM, the minute hand is at 12, and the hour hand is at 7. Then, the angle is not yet  ( really), and Tammy is still exercising.
The minute hand will keep getting ahead, but after t is ahead, and if it gained another it would catch up with the hour hand again.
After that the angle gets smaller until it is , and Tammy stops exercising.
During Tammy's exercise, the minute hand went from a position (or 15 minutes) behind the hour hand to a position (or 20 minutes) behind the hour hand, lapping it once.
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