To determine how many times the hands of a clock are at a $0^\circ$ angle (overlap) between Monday 7:00 am and Tuesday 9:00 pm, we can break the problem into manageable steps.

Key Observations:

• The hands of a clock overlap every 12/11 hours.
• In a full 12-hour period, the hands overlap 11 times.

Time Span:

1. From Monday 7:00 am to Tuesday 9:00 pm, we have 38 hours:

   • From Monday 7:00 am to Monday 7:00 pm: 12 hours.
   • From Monday 7:00 pm to Tuesday 7:00 pm: 24 hours.
   • From Tuesday 7:00 pm to Tuesday 9:00 pm: 2 hours.

   So, total hours = $12 + 24 + 2 = 38$ hours.

Calculation:

• In a 12-hour period, there are 11 overlaps.
• In a 24-hour period (full day), there are 22 overlaps.
• For the 2 additional hours on Tuesday, there is 1 overlap, because overlaps occur approximately every 1 hour and 5 minutes ($12/11 \approx 1.09$).

Total Overlaps:

• From Monday 7:00 am to Tuesday 7:00 pm (36 hours): $33$ overlaps (since $11 \times 3 = 33$).
• Plus 1 overlap between Tuesday 7:00 pm and Tuesday 9:00 pm.

So, the total number of overlaps is $33 + 1 = 34$.

Final Answer:

The hands of the clock overlap 34 times between Monday 7:00 am and Tuesday 9:00 pm.

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Would you like to dive into more details about the process? Here are some related questions:

1. How do we calculate when the hands of a clock form a 180° angle?
2. Can the overlap times be predicted for any given hour?
3. How does the clock angle formula work for other angles?
4. What is the derivation for the formula $12/11$ hours for overlap?
5. How does the time between overlaps change if the clock speed is altered?

Tip: The $12/11$ hour interval comes from the fact that the minute hand moves 12 times faster than the hour hand, and they meet roughly every 65.45 minutes.