Concept:
The hands of a clock are opposite to each other when the angle between them is 1800.
The relative speed of the minute hand with respect to the hour hand is 5.50 per minute.
The hour hand covers 3600 in 12 hours, so in 1 hour it covers 300.
Calculation:
At 4 AM, the minute hand is at 12 and the hour hand is at 4.
Angle between the hands at 4:00 = 4 × 300 = 1200.
To be opposite (1800), the minute hand must cover the initial 1200 gap and then gain an additional 1800.
Total relative angular distance to be covered = 1200 + 1800 = 3000.
Time taken = Distance ÷ Relative Speed
Time = 300 ÷ 5.5 = 600 ÷ 11
Time ≈ 54.54 minutes.
The approximate time is 4:54.54 AM, which rounds to 4:55 AM.
Hence, the correct answer is “Option 1”.
Alternate Method
The angle between the hour hand and minute hand is given by the formula: θ = |30H − 5.5M|
Here, θ = 1800 (opposite hands) and H = 4.
180 = |30 × 4 − 5.5M|
180 = |120 − 5.5M|
Since the minute hand has passed the hour hand to be opposite, we use:
180 = 5.5M − 120
180 + 120 = 5.5M
300 = 5.5M
M = 300 ÷ 5.5 = 600 ÷ 11 ≈ 54.54
The time is approximately 4:55 AM.
Hence, the correct answer is 4:55.