Concept:
The hour hand covers 360° in 12 hours.
Angular distance covered by the hour hand in 1 minute = 360 ÷ (12 × 60) = 0.5°
The minute hand covers 360° in 60 minutes.
Angular distance covered by the minute hand in 1 minute = 360 ÷ 60 = 6°
The relative speed between the two hands = 6° − 0.5° = 5.5° per minute.
For the hands of a clock to be opposite to each other, the angle between them must be 180°.
Calculation:
At 3 PM, the minute hand is at 12 (0°) and the hour hand is at 3 (90°).
To be opposite, the minute hand must first cover the 90° gap to meet the hour hand and then move 180° further to be exactly opposite.
Total relative angular distance to be covered = 90° + 180° = 270°
Time taken = Distance ÷ Relative speed
Time = 270 ÷ 5.5 = 270 ÷ (11/2)
Time = 540/11 minutes ≈ 49.09 minutes
The time is approximately 49 minutes past 3, which is 3:49 PM.
Alternate Method
The angle between the hour hand and minute hand is given by the formula:
Angle = |30H − (11/2)M|
Where H is hours and M is minutes.
Here, H = 3 and the hands are opposite, so Angle = 180°.
180 = |30 × 3 − 5.5 × M|
180 = |90 − 5.5M|
Since the minute hand is ahead of the hour hand to form a 180° angle after 3 PM:
180 = 5.5M − 90
180 + 90 = 5.5M
270 = 5.5M
M = 270 ÷ 5.5 = 540/11
M ≈ 49.09 minutes
Hence, the correct answer is 3:49.