Correct Option (C)
To determine the time when the minute hand is 3 minutes ahead of the hour hand between 6 p.m. and 7 p.m., we calculate the relative positions and speeds of the clock hands.
- The minute hand moves at 6 degrees per minute.
- The hour hand moves at 0.5 degrees per minute.
- The relative speed of the minute hand with respect to the hour hand is 5.5 degrees per minute (6 - 0.5).
At 6:00 p.m., the hour hand is at the 6-mark (180 degrees from the 12-mark), and the minute hand is at the 12-mark (0 degrees). The hour hand is initially 180 degrees ahead of the minute hand.
For the minute hand to be 3 minutes ahead of the hour hand, it needs to cover the initial 180-degree gap and then move an additional 3 minutes (which is 3 × 6 = 18 degrees) ahead of the hour hand's current position.
Let 't' be the number of minutes past 6:00 p.m.
- Angle of the minute hand from 12: 6t degrees.
- Angle of the hour hand from 12: 180 degrees (initial position at 6) + 0.5t degrees (movement in 't' minutes).
We set up the equation where the minute hand's angle is 18 degrees greater than the hour hand's angle:
6t = (180 + 0.5t) + 18
6t = 198 + 0.5t
5.5t = 198
t = 198 / 5.5
t = 36 minutes
Therefore, at 6:36 p.m., the minute hand will be 3 minutes ahead of the hour hand.
Incorrect Options:
The positions of the hands for the other options do not satisfy the given condition:
- 6:15 p.m.:
- Minute hand is at 15 minutes (90 degrees from 12).
- Hour hand is at 6 + 15/60 = 6.25. Angle = 6.25 × 30 = 187.5 degrees from 12.
- The minute hand (90 degrees) is behind the hour hand (187.5 degrees).
- 6:18 p.m.:
- Minute hand is at 18 minutes (108 degrees from 12).
- Hour hand is at 6 + 18/60 = 6.3. Angle = 6.3 × 30 = 189 degrees from 12.
- The minute hand (108 degrees) is behind the hour hand (189 degrees).
- 6:48 p.m.:
- Minute hand is at 48 minutes (288 degrees from 12).
- Hour hand is at 6 + 48/60 = 6.8. Angle = 6.8 × 30 = 204 degrees from 12.
- The minute hand is ahead of the hour hand by 288 - 204 = 84 degrees. This angular difference corresponds to 84/6 = 14 minutes, not 3 minutes.
Thus, options A, B, and D are incorrect as they do not correspond to the minute hand being 3 minutes ahead of the hour hand.