The given time interval is between 8 AM and 9 AM.
At 8:00 AM, the hour hand is at 8 × 300 = 2400 and the minute hand is at 00.
To find the time when the angle is 75 degrees, we consider the relative speed = 6 − 0.5 = 5.50 per minute.
Case 1: The minute hand is behind the hour hand.
Relative angular distance to be covered = 240 − 75 = 1650
Time = 165 / 5.5 = 30 minutes. (This corresponds to 8:30 AM)
Case 2: The minute hand has crossed the hour hand.
Relative angular distance to be covered = 240 + 75 = 3150
Time = 315 / 5.5 = 630 / 11 ≈ 57.27 minutes.
This corresponds approximately to 8:57 AM.
Hence, the correct answer is “Option 3”.
Alternate Method
The angle between the hour hand and minute hand = \(|60H - 11M|\over2\)
Where H = hours and M = minutes.
Given Angle = 75 and H = 8.
75 = \(|60 \times 8 - 11M|\over2\)
150 = |480 − 11M|
480 − 11M = −150 (since we are looking for the time closer to 9 AM based on options)
11M = 480 + 150
11M = 630
M = 630/11 ≈ 57.27
Therefore, the time is approximately 8:57 AM.
Hence, the correct answer is 8:57.