Final answer:
To determine the current time when it is known that the minute and hour hands of a clock will reverse their relative positions in 20 minutes, one must calculate the movement of both hands and solve the resulting equation. In this case, the time is determined to be 3:10.
Step-by-step explanation:
The question involves finding the current time given that it is between 3 and 4 o'clock and that in 20 minutes, the minute hand will have moved such that it is now as far ahead of the hour hand as it was previously behind it.
To solve this, we must understand the mechanics of a clock's hands.
The hour hand moves 0.5 degrees per minute (360 degrees divided by 12 hours divided by 60 minutes), and the minute hand moves 6 degrees per minute (360 degrees divided by 60 minutes).
When it's exactly 3 o'clock, the hour hand is at 90 degrees (3 hours multiplied by 30 degrees per hour).
After t minutes, the hour hand would have moved 0.5t degrees from that 90-degree mark, and the minute hand would be at 6t degrees from the noon position.
The original angle between the hands was therefore 90 degrees minus 0.5t, and in 20 minutes, the angle will be 6(t+20) minus 90 minus 0.5t degrees.
Setting up the equation: 90 - 0.5t = 6(t+20) - 90 - 0.5t, which simplifies to 180 = 6(t+20).
Solving this gives us t = 10 minutes.
Hence, the current time is 10 minutes past 3 o'clock, or 3:10.