Step 1: Problem Definition:
This problem involves a malfunctioning clock that consistently loses time. The objective is to determine the actual elapsed time when the faulty clock displays a specific duration.
Step 2: Methodology:
1. Calculate the total duration indicated by the faulty clock.
2. Establish the correlation between the faulty clock's readings and the actual time.
3. Compute the true elapsed time.
4. Ascertain the correct time.
Step 3: Calculation Details:
The clock was set at 5 a.m. on Day 1. We need to find the correct time when the faulty clock indicates 10 p.m. on Day 4.
Faulty clock's elapsed time:
From Day 1, 5 a.m. to Day 4, 5 a.m. = 3 days = \(3 \times 24 = 72\) hours.
From Day 4, 5 a.m. to Day 4, 10 p.m. = 17 hours.
Total time displayed by the faulty clock = \(72 + 17 = 89\) hours.
Rate of time loss: The clock loses 16 minutes per 24 hours.
This implies that for every 24 hours of correct time, the faulty clock shows 23 hours and 44 minutes.
23 hours 44 minutes = \(23 \frac{44}{60}\) hours = \(23 \frac{11}{15}\) hours = \(\frac{356}{15}\) hours.
Therefore, \(\frac{356}{15}\) hours on the faulty clock corresponds to 24 hours of correct time.
Consequently, 1 hour on the faulty clock equals \(24 \times \frac{15}{356}\) hours of correct time, which simplifies to \(\frac{90}{89}\) hours of correct time.
With the faulty clock showing 89 hours, the correct elapsed time is calculated as:
Correct time = \(89 \times (\text{correct time per faulty hour})\)
Correct time = \(89 \times \frac{90}{89} = 90\) hours.
The actual time elapsed is 90 hours from the start time (Day 1, 5 a.m.).
90 hours is equivalent to 3 days and 18 hours (\(90 = 3 \times 24 + 18\)).
Start time: Day 1, 5 a.m.
After 3 days, the time is Day 4, 5 a.m.
Adding the remaining 18 hours to Day 4, 5 a.m.:
5 a.m. + 18 hours = 23:00, which is 11 p.m.
Step 4: Conclusion:
The correct time is 11 p.m. on the 4th day.