SOLUTION: there is a clock. The long and short hands are on a straight line. How long does it take for the 2 hands to overlap.

Question 1163942: there is a clock. The long and short hands are on a straight line. How long does it take for the 2 hands to overlap.
Answer by ikleyn(52866) (Show Source): You can put this solution on YOUR website!
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there is a clock. The long and short hands are on a straight line .
How long does it take for the 2 hands to overlap.
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It is clear that the answer is the same for any possible positions of the two hands satisfying the given condition.


Therefore, let's assume that the initial position of the hands is 6:00 PM.


The minute hand is in position 90° vertically up.
The hour hand is in position -90° vertically down.


The angular speed of the minute hand is 360° per hour, or   = 6 degrees per minute.

The angular speed of the hour   hand is 360° per 12 hours, or   =  = 0.5 degrees per minute.


The position of the minute hand t minutes after 6:00 pm is  90 - 6t  degrees.

The position of the hour   hand t minutes after 6:00 pm is  -90 - 0.5t  degrees.


The hands overlap means

    90 - 6t = -90 - 0.5t,   or

    90 + 90 = 6t - 0.5t

    180     = 5.5t

    t       =  =  = 32.7272... minutes = 32 minutes 43.632 seconds, or approximately 32 minutes and 44 seconds.


So, the answer to the problem's question is   minutes, or 32 minutes and 44 seconds.

Solved.

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To see many other similar solved problems, look into the lessons
- Clock problems
- Advanced clock problems
in this site.

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