Question 1198138
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Between 6 p.m. and 7 p.m. the hands of a clock make a ninety-degree angle on two occasions. 
If Jenny always leaves home to walk her dog when the first ninety-degree angle is formed, 
and arrives home when the second is formed, how much time, in hours, 
does Jenny spend walking her dog every week?
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<pre>
We will measure angles starting from the position of the hour and the minute hands vertically up, at 12:00 (midday, the noon).


The minute hand makes one full rotation in one hour, so its angular velocity is 360 degrees per hour, or  

    360/60 = 6 degrees per minute.


The hour hand makes one full rotation in 12 hours, so its angular velocity is 360 degrees per 12 hours, or

    360/12 = 30 degrees per hour = 30/60 = 0.5 degrees per minute.



At 6:00 pm, the hour hand   is in position 6*30 = 180 degrees from vertical position clockwise.

            The minute hand is in vertical position (= 0 degrees) at that time.


"t" minutes after 6:00 pm, the hour hand is in position 180 + 0.5t  degrees from vertical up;

                           the minute hand is in position 6t  degrees.



After 6:00 pm, the hour hand and the minute hand make the right angle for the first time, when  180 + 0.5t = 6t + 90  degrees.

From this equation, we find  

     180 - 90 = 6t - 0.5t,  or  5.5t = 90,  t = {{{90/5.5}}} = 16.3636... minutes.



After 6:00 pm, the hour hand and the minute hand make the right angle for the second time, when  6t = 180 + 0.5*t + 90  degrees.

From this equation, we find  

     6t - 0.5t = 180 + 90,  or  5.5t = 270,  t = {{{270/5.5}}} = 49.0909... minutes.


<U>ANSWER</U>.  Ian was out with his dog 49.0909 - 16.3636 = 32.7272 minutes = 32 minutes and 44 seconds (approximately.
</pre>

Solved.


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On clock problems, see my 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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