Question 1092284
.
<pre>
The way to solve this problem is to write equation for angles that the minute hand and the hour hand make with . . . let say, 
with the vertical direction, with the direction "vertically up".

So, we will count angles clockwise starting from that direction "vertically up".


The minute hand makes one full revolution in 1 hour = 60 minutes. Hence, its angular velocity is  {{{360/60}}} = 6 degrees per minute.

The hour hand makes one full revolution in 12 hours. Hence, its angular velocity is 12 times less, i.e. {{{(1/12)*6}}} = {{{1/2}}} degrees per minute.


Let "t" be the number of minutes after 2:00 pm which is under the question.
Now the minute hand is in position 6t degrees, while the hour hand is in position  {{{60 + (6t)/12}}} degrees.
Thus the angle between the hands is  {{{60 + (6t)/12 - 6t}}} degrees.  

      Notice that 60 = 60 degs corresponds to the position of the hour hand at 2:00 pm.


In 10 minutes, the minute hand will be in position of  6(t+10) degrees, while the hour hand will be in position of {{{60 + (6*(t+10))/12}}} degrees; 
thus the angle between the hands will be {{{6(t+10) - (60 + (6*(t+10))/12)}}}.


We should determine the time moment "t" by equating these two angles:

{{{6(t+10) - (60 + (6*(t+10))/12)}}} = {{{60 + (6t)/12 - 6t}}}.   (1)


To solve the equation (1), multiply both sides by 12. You will get

12*6*(t+10) - 60*12 - 6*(t+10) = 60*12 + 6t - 12*6t.


Simplify:

12*6*t + 12*6*10 - 60*12 - 6t - 60 = 60*12 + 6t - 12*6t,

12*6*t - 6t - 6t + 12*6*t = 60*12 - 12*6*10 + 60*12 + 60,

132t = 780  ====>  t = {{{780/132}}} = 5.909 minutes = 5 minutes 54.(54) seconds = 5 minute and 54.546 seconds (approximately). 
</pre>

<U>Answer</U>.  The time under the question is 2 hours 5 minutes and 54.546 seconds (approximately; correct up-to 3 decimals after the decimal dot).



Solved.