SOLUTION: what is the time between 2 and 3 o'clock when the hands of the clock form an angle 204 degrees with each other. find time between 2 and 3.

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Question 939418: what is the time between 2 and 3 o'clock when the hands of the clock form an angle 204 degrees with each other. find time between 2 and 3.
Found 2 solutions by josmiceli, Edwin McCravy:
Answer by josmiceli(19441) About Me  (Show Source):
You can put this solution on YOUR website!
Divide the clock into +360+ degrees
In one hour, the hour hand moves
+%285%2F60%29%2A360+=+30+ degrees
So, it's rate is +30+ degrees/hr
-------------------------------
In one hr, the minute hand moves +360%2F1+
degrees/hr
--------------
At exactly 2:00, the hands are +%2810%2F60%29%2A360+=+60+
degrees apart with the minute hand behind
--------------
If +t+ = fraction of 1 hr
The hour hand moves +30t+
The minute hand moves +-60+%2B+360t+
+-60+%2B+360t+-+30t+=+204+
+330t+=+264+
+t+=+.8++
+.8%2A60+=+48+
The time would be 2:48
( definitely get another opinion on
this. I could easily mess this up! )


Answer by Edwin McCravy(20056) About Me  (Show Source):
You can put this solution on YOUR website!
The other tutor's answer is correct.  I'll use a
slightly different approach and a little more detail.

Minute hand's rate=          |  Hour hand's rate=     
360 degrees per 1 hour=      | 360 degrees per 12 hours= 
360 degrees per 60 minutes=  |  30 degrees per 1 hour= 
6 degrees per minutes        |  30 degrees per 60 minutes=
                             | 1/2 degree per minute

At 2:00 the minute hand is straight up which is 60 degrees 
behind the hour hand, which is pointing at the numeral 2.
So the minute hand mst catch up to the hour hand and pass it
until it is by 204 degrees ahead of the hour hand.

Let the answer be x minutes after 2:00

In x minutes the hour hand has turned through (1/2)x degrees.
In those x minutes the minute hand has turned through 6x degrees 

To be 240 degrees ahead of the hour hand in x minutes, the minute
hand must have turned through these angles:

1. the number of degrees it was behind the hour hand at 2:00, which
   is 60 degrees.

plus

2. the number of degrees which the hour hand has turned through in 
   x minutes since 2:00, which is (1/2)x degrees.

plus

3. 204 more degrees, in order to be that many degrees ahead of the 
   hour hand. 

That must total up to the number of degrees the minute hand has turned,
which is 6x degrees.

So the equation is

         60 + (1/2)x + 204 = 6x
              (1/2)x + 264 = 6x
Multiply through by 2
                   x + 528 = 12x
                       528 = 11x
                        48 = x

So the answer is 2:48.

Edwin