Question 1163942
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there is a clock. The long and short hands are on a straight line {{{highlight(in_opposite_directions)}}}. 
How long does it take for the 2 hands to overlap.
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<pre>
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  {{{360/60}}} = 6 degrees per minute.

The angular speed of the hour   hand is 360° per 12 hours, or  {{{360/(12*60)}}} = {{{6/12}}} = 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       = {{{180/5.5}}} = {{{360/11}}} = 32.7272... minutes = 32 minutes 43.632 seconds, or approximately 32 minutes and 44 seconds.


So, the answer to the problem's question is  {{{360/11}}} minutes, or 32 minutes and 44 seconds.
</pre>

Solved.


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To see many other similar solved problems, look into the lessons

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/word/travel/Clock-problems.lesson>Clock problems</A> 

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/word/travel/Advanced-clock-problems.lesson>Advanced clock problems</A>

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