SOLUTION: Tammy starts her aerobic exercises every day after 6 a.m., when the hands of the clock make a 90° angle. She finishes after 7 a.m., when the hands of the clock make a 120° angle.

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Question 1168688: Tammy starts her aerobic exercises every day after 6 a.m., when the hands of the clock make a 90° angle. She finishes after 7 a.m., when the hands of the clock make a 120° angle. How much time does Tammy spend exercising every week?
Answer by VFBundy(438) About Me  (Show Source):
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Think of the clock as a 360 degree circle where 0 degrees is at the '12', 90 degrees at the '3', and so on and so forth.

Degree position of minute hand at 6 AM = 0
Degree position of hour hand at 6 AM = 180

We want to figure out the first time after 6 AM where those two hands are 90 degrees apart.

It takes the minute hand one hour to make one complete revolution around the clock. This means, each minute, the minute hand moves 1/60 of the way around the clock. Since the clock is 360 degrees around, that means the minute hand moves 6 degrees every minute. (1/60 * 360 degrees = 6 degrees.)

The hour hand moves 1/12 times slower than the minute hand. (We know this because it takes 12 hours for the hour hand to make one complete revolution around the clock, as opposed to just one hour for the minute hand.) Since we now know the minute hand moves 6 degrees per minute, this means we can compute that the hour hand moves 0.5 degrees per minute. (1/12 * 6 degrees = 1/2 degrees...or, 0.5 degrees.)

Let x be the number of minutes after 6 AM that it takes for the two hands to be exactly 90 degrees apart.

(180 + 0.5x) - (0 + 6x) = 90

180 + 0.5x - 0 - 6x = 90

180 - 5.5x = 90

-5.5x = -90

5.5x = 90

x = 16.36

This means Tammy starts exercising 16.36 minutes after 6 AM.

For the second part of the problem, we basically do the same thing, except that Tammy stops exercising after 7 AM the first time the clock's hands are 120 degrees apart.

Degree position of minute hand at 7 AM = 0
Degree position of hour hand at 7 AM = 210 (360 * 7/12)

This time, let x be the number of minutes after 7 AM that it takes for the two hands to be exactly 120 degrees apart.

(210 + 0.5x) - (0 + 6x) = 120

210 + 0.5x - 0 - 6x = 120

210 - 5.5x = 120

-5.5x = -90

5.5x = 90

x = 16.36

As you can see, we get the same exact answer as we did in the first part of the problem. Tammy stops exercising at 16.36 minutes after 7 AM.

So, Tammy exercises exactly one hour per day. (From 16.36 minutes after 6 AM until 16.36 minutes after 7 AM.) This means Tammy exercises exactly SEVEN HOURS per week.