To determine the actual time when the clock indicates 10 p.m., considering it gains 2 minutes every hour from its setting at 6 a.m.
- Initial Setting: 6 a.m.
- Rate of Gain: 2 minutes per hour.
- Observed Time: 10 p.m.
- Actual Time: The real elapsed time when the malfunctioning clock displays 10 p.m.
- The clock advances by 2 minutes each hour.
- The duration recorded by the faulty clock from 6 a.m. to 10 p.m. is 16 hours.
For every hour of true time, the clock advances by \( 60 + 2 = 62 \) minutes.
Let \( t \) represent the true elapsed time in hours.
The time displayed by the clock after \( t \) hours is calculated as:
\( \text{Clock Display} = \frac{62}{60} \times t = \frac{31}{30} t \)
Given that the clock displays 16 hours (from 6 a.m. to 10 p.m.):
\[ \frac{31}{30} t = 16 \]
Solving for \( t \):
\[ t = \frac{16 \times 30}{31} = \frac{480}{31} \approx 15.48 \text{ hours} \]
The fractional part of the hours is 0.48 hours, which converts to minutes as:
0.48 hours =\( 0.48 \times 60 = 28.8 \text{ minutes} \), approximating to 29 minutes.
More precisely, \( \frac{480}{31} \) hours is equivalent to:
Full Hours = 15 hours
Remaining Minutes = \( \frac{480}{31} - 15 = \frac{480 - 465}{31} = \frac{15}{31} \) hours
Converting these remaining hours to minutes:
\( \frac{15}{31} \times 60 \approx 29.03 \) minutes. This value is closer to 29 minutes. For exact calculation, the minutes are 28.8, which translates to 28 minutes and 48 seconds.
Starting from the set time of 6 a.m., adding the calculated true elapsed time of approximately 15 hours and 28 minutes yields:
6:00 a.m. + 15 hours 28 minutes = 9:28 p.m. (after accounting for the 12-hour cycle).
The actual time when the clock shows 10 p.m. is approximately 9:28 p.m.