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# help

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The time is 9:00. On a 12-hour clock, how many minutes elapse before the hour hand and minute hand are 28 degrees apart for the first time?

Nov 3, 2019

#1
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At 9:00, the hands form a right-angle (90°) or you can think of it as 270° if you look at the angle between the minute and the hour hand.

MINUTE HAND RATE:

Now the minute hand moves 1/60 of the way around the circle each minute. 1/60 of 360° is 6°

The minute hand moves 6° per minute.

HOUR HAND RATE:

The hour hand moves 1/12 of the way each hour. Dividing by 60, that means the hour hand moves 1/720 of the way around the circle each minute. 1/720 * 360° = ½°

The hour hand moves ½° per minute.

Thus the minute hand is getting 6° closer, but the hour hand is moving away at a rate of ½° per minute. That's a net of 5½° per minute that they are getting closer together.

You want to figure the minutes until the 270° angle is down to 28°.

Set up an equation:

270 - 5.5t = 28

270 - 28 = 5.5t

242 = 5.5t

t = 242/5.5

t = 484/11

t = 44

In 44 minutes (at 9:44).

Nov 3, 2019

#1
+2

At 9:00, the hands form a right-angle (90°) or you can think of it as 270° if you look at the angle between the minute and the hour hand.

MINUTE HAND RATE:

Now the minute hand moves 1/60 of the way around the circle each minute. 1/60 of 360° is 6°

The minute hand moves 6° per minute.

HOUR HAND RATE:

The hour hand moves 1/12 of the way each hour. Dividing by 60, that means the hour hand moves 1/720 of the way around the circle each minute. 1/720 * 360° = ½°

The hour hand moves ½° per minute.

Thus the minute hand is getting 6° closer, but the hour hand is moving away at a rate of ½° per minute. That's a net of 5½° per minute that they are getting closer together.

You want to figure the minutes until the 270° angle is down to 28°.

Set up an equation:

270 - 5.5t = 28

270 - 28 = 5.5t

242 = 5.5t

t = 242/5.5

t = 484/11

t = 44