If it is now 3 o'clock, at what time will the hands of the clock be on the same line for the first time?

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

Discover how to calculate the first time the clock hands will align after 3 o'clock. This problem covers key time and geometry concepts such as relative speed, time calculations, and linear motion in circular geometry. Ideal for students in Grades 9-11.

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