Short Answer 
The minute and hour hands of the clock will meet approximately 62 minutes after 3:20, which occurs around 4:16 AM. The calculations involved determining their positions in degrees and the relative speed at which they move.
Step 1: Determine the Position of the Minute Hand
At 3:20, the minute hand is located at the number 4 on the clock, indicating that it is 20 minutes past the hour. To find its position in degrees, we know that the minute hand moves at a rate of 6 degrees per minute. Therefore, by calculating:
- 20 minutes ‚à öo 6 degrees/minute = 120 degrees
This shows that the minute hand is positioned at 120 degrees from the 12 o’clock position.
Step 2: Determine the Position of the Hour Hand
For the hour hand, it moves at 30 degrees per hour. At 3:00, it is at 90 degrees. Between 3:00 and 3:20, the hour hand moves slightly further:
- 20 minutes is one-third of an hour.
- 30 degrees/hour ‚à öo 1/3 = 10 degrees
Thus, at 3:20, the hour hand is positioned at 100 degrees.
Step 3: Calculate the Time Until the Hands Meet
To find when the minute hand will catch up with the hour hand, we first determine the relative speed of their movements. The minute hand moves at 6 degrees per minute, while the hour hand moves at 0.5 degrees per minute:
- Relative speed = 6 degrees/minute – 0.5 degrees/minute = 5.5 degrees/minute
The minute hand needs to travel a total of 340 degrees to catch up with the hour hand. This gives us the equation:
- Time = 340 degrees / 5.5 degrees/minute
Solving this results in approximately 61.8 minutes, which rounds to about 62 minutes. Hence, the hands will meet at around 4:16 AM.