To find the first time the hands of the clock will be on the same line after 3 o'clock, let's break down the problem.

Step 1: Understanding the movement of the clock hands

• Hour hand: The hour hand moves at a rate of 0.5 degrees per minute (360 degrees/12 hours = 30 degrees per hour, and 30 degrees/60 minutes = 0.5 degrees per minute).
• Minute hand: The minute hand moves at a rate of 6 degrees per minute (360 degrees/60 minutes = 6 degrees per minute).

Step 2: Initial positions of the hands at 3 o'clock

• At exactly 3 o'clock, the hour hand is at the 90-degree position (3 hours × 30 degrees per hour = 90 degrees).
• The minute hand is at 0 degrees (since it points to the 12 at the top of the clock).

Step 3: Relative speed of the two hands

The minute hand moves faster than the hour hand. The relative speed between the minute hand and the hour hand is: $\text{Relative speed} = 6 \, \text{degrees/min} - 0.5 \, \text{degrees/min} = 5.5 \, \text{degrees/min}.$

Step 4: When will the hands align?

For the hands to be on the same line again, the minute hand needs to "catch up" to the hour hand, covering the initial 90-degree gap. The time taken for this to happen is: $\text{Time} = \frac{\text{Initial gap}}{\text{Relative speed}} = \frac{90^\circ}{5.5 \, \text{degrees/min}} \approx 16.36 \, \text{minutes}.$

Step 5: Conclusion

The hands will be aligned (on the same line) after approximately 16.36 minutes. Therefore, the first time they will be on the same line after 3 o'clock is around 3:16.36 (or 3:16 and 22 seconds).

Would you like a more detailed explanation or any clarification?

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Here are 5 related questions you might find interesting:

1. How often do the clock hands align in 12 hours?
2. How can we calculate the time when the hands form a 90-degree angle after 3 o'clock?
3. How long after 3 o'clock will the hands overlap completely (one on top of the other)?
4. What is the next time after 3 o'clock that the hands will be exactly opposite each other?
5. How can we determine when the hands are at a specific angle, like 60 degrees apart?

Tip: Remember, the relative speed between