The grandfather clock at the top of the stairs says it is now 3:20AM. In how many minutes from now will the hour and minute hand of the clock first meet each other? Please round to the nearest minute.

To solve this problem, we need to determine the time it will take for the hour and minute hands of the clock to first meet after 3:20 AM. 

### Step 1: Convert the current time to angles
- **Hour hand position**: At 3:20 AM, the hour hand is not exactly on 3, since it has moved due to the 20 minutes that have passed.
  - Each hour represents 30 degrees (360 degrees / 12 hours).
  - So, the hour hand at 3:00 is at \(3 \times 30 = 90\) degrees.
  - In 20 minutes, the hour hand moves \( \frac{20}{60} \times 30 = 10 \) degrees.
  - Therefore, the hour hand is at \(90 + 10 = 100\) degrees.

- **Minute hand position**: At 20 minutes past the hour, the minute hand is at \(20 \times 6 = 120\) degrees (since each minute represents 6 degrees).

### Step 2: Calculate the relative speed of the hands
- **Minute hand speed**: The minute hand moves 6 degrees per minute.
- **Hour hand speed**: The hour hand moves 0.5 degrees per minute.

The relative speed at which the minute hand approaches the hour hand is:
\[
6 - 0.5 = 5.5 \text{ degrees per minute}
\]

### Step 3: Determine the initial angular difference
The initial difference in position between the minute hand and the hour hand is:
\[
120^\circ - 100^\circ = 20^\circ
\]

### Step 4: Time to meet
To find when they will meet, we calculate how long it will take for the 20-degree difference to be closed at the relative speed of 5.5 degrees per minute:
\[
\text{Time} = \frac{20^\circ}{5.5^\circ/\text{minute}} \approx 3.636 \text{ minutes}
\]

Rounding to the nearest minute, the hour and minute hands will first meet in **4 minutes**.

### Conclusion
The hour and minute hands of the clock will first meet each other approximately **4 minutes** from now.

---

Would you like further details on any part of this solution? Here are five related questions you might find interesting:
1. How does the calculation change if the clock reads 4:45 AM instead of 3:20 AM?
2. What would be the meeting time if the hands were exactly 180 degrees apart initially?
3. Can you determine how many times the hour and minute hands meet within 12 hours?
4. How do the speeds of the hour and minute hands compare over the course of a full day?
5. What is the angle between the hands at 9:45 AM?

**Tip**: Understanding the relative speeds of clock hands can help solve many types of clock-related problems.

Learn how to calculate when the hour and minute hands of a clock will first meet at 3:20 AM with a step-by-step solution. This problem involves geometry, trigonometry, and clock arithmetic, analyzing the angular speeds of clock hands. Suitable for advanced high school students.

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