To solve this problem, we need to calculate when the minute and hour hands of a clock will next overlap after 9:10 a.m.

Step-by-Step Solution

1. Understanding Initial Positions:

   • At 9:00, the hour hand points at 9, which is 270 degrees from 12 o'clock (each hour is 30 degrees).
   • At 9:10, the minute hand is on 2 (10 minutes into the hour), which is 60 degrees (each minute moves the minute hand by 6 degrees).

2. Calculating the Position of the Hour Hand at 9:10:

   • Every hour, the hour hand moves 30 degrees (360 degrees / 12 hours).
   • In 10 minutes (1/6 of an hour), the hour hand moves $\frac{30}{6} = 5$ degrees.
   • Thus, at 9:10, the hour hand is at $270 + 5 = 275$ degrees from the 12 o'clock position.

3. Relative Speed of Minute and Hour Hands:

   • The minute hand moves at 6 degrees per minute.
   • The hour hand moves at 0.5 degrees per minute.
   • Therefore, the relative speed at which the minute hand approaches the hour hand is $6 - 0.5 = 5.5$ degrees per minute.

4. Distance to Close the Gap:

   • At 9:10, the gap between the minute and hour hands is $360 - (275 - 60) = 145$ degrees.
   • To close this gap, we calculate the time it will take: $\text{Time} = \frac{\text{Gap}}{\text{Relative Speed}} = \frac{145}{5.5} \approx 26.36 \text{ minutes}$

5. Convert Minutes to Seconds:

   • $26.36$ minutes is approximately $26.36 \times 60 \approx 1582$ seconds.

6. Conclusion:

   • Rounded to the nearest second, it will take approximately 1582 seconds after 9:10 for the hour and minute hands to overlap.

Would you like a more detailed breakdown of any step?

Here are five questions to further explore this topic:

1. How often do the minute and hour hands overlap in a 12-hour period?
2. How would the calculation change if the time was 9:20 instead of 9:10?
3. What is the angle between the hour and minute hands at any given time?
4. At what other times do the hour and minute hands overlap throughout the day?
5. How would we find when the minute and hour hands overlap on a 24-hour clock?

Tip: Remember that the speed difference between the hands is key in solving overlap problems.