To find the first time after 2 o'clock when the hands of a clock are 120 degrees apart, we can use the following method.

Step-by-Step Solution:

1. Basic Information:

   • The hour hand moves 0.5 degrees per minute (360 degrees / 12 hours / 60 minutes = 0.5 degrees per minute).
   • The minute hand moves 6 degrees per minute (360 degrees / 60 minutes = 6 degrees per minute).

2. Position of the hands at 2:00:

   • At exactly 2:00, the hour hand is at 60 degrees (since each hour represents 30 degrees and 2 hours = 2 × 30 = 60 degrees).
   • The minute hand is at 0 degrees (aligned with the 12 o'clock position).

3. Setting up the equation: Let $t$ represent the time in minutes after 2:00.

   • The position of the hour hand after $t$ minutes is: $60 + 0.5t \text{ degrees}.$
   • The position of the minute hand after $t$ minutes is: $6t \text{ degrees}.$ We need the difference between these two positions to be 120 degrees, so the equation becomes: $|(60 + 0.5t) - 6t| = 120.$ This can be simplified into two possible equations (since the absolute value gives two scenarios):

4. Solving the equations:

   • First case: $6t - (60 + 0.5t) = 120$ $6t - 60 - 0.5t = 120$ $5.5t = 180$ $t = \frac{180}{5.5} = 32.73 \text{ minutes}.$ So, the first solution is approximately 32.73 minutes after 2:00, or 2:32:44.

   • Second case: $(60 + 0.5t) - 6t = 120$ $60 + 0.5t - 6t = 120$ $-5.5t = 60$ $t = \frac{60}{-5.5} = -10.91 \text{ minutes}.$ This is not valid since we're looking for the time after 2:00.

Final Answer:

The first time after 2 o'clock when the hands of the clock are 120 degrees apart is approximately 2:32:44.

Do you want further details or have any questions? Here are some related questions:

1. What is the formula to find the angle between clock hands at any time?
2. How often are the hands of a clock 120 degrees apart in a 12-hour period?
3. Can we use a graphical method to solve the same problem?
4. What is the relationship between the speeds of the hour and minute hands?
5. How do we calculate other specific angles between clock hands?

Tip: You can generalize this approach to find any angle between clock hands at any time!