The light in a restroom operates with a 15-minute timer that is reset every time the door opens as a person goes in or out of the room. Thus, after someone enters or exits the room, the light remains on for only 15 minutes unless the door opens again and reset the timer for another 15 minutes. If the times listed above are the times at which the door opened from 8:00 to 10:00, approximately how many minutes during this two-hour period was the light off? Times at Which the Door Opened from 8:00 to 10:00
Show Hint
To solve this type of problem, first, sort all the time points. Then, go through the list and find any two consecutive times that are more than 15 minutes apart. The "off" time is the difference between these two times, minus the 15-minute timer duration. For the last entry, calculate the off-time until the end of the total period.
Step 1: Understanding the Concept:
We need to calculate the total duration the light was off in a 120-minute period (from 8:00 to 10:00). The light turns on at the first door opening and stays on for 15 minutes from the *last* door opening in any continuous series of events. An "off" period occurs when more than 15 minutes pass between two consecutive door openings. Step 2: Detailed Explanation:
Let's trace the state of the light. An "ON" period ends 15 minutes after the last door opening in a cluster. The light turns off only if the next opening is after this "turn-off" time.
ON Period 1: The first opening is at 8:00. The light turns on. We look for a gap of more than 15 minutes between subsequent openings. The sequence of openings is 8:00, 8:03, 8:04, 8:06, 8:10, 8:18, 8:19, 8:30, 8:31. The gap between 8:10 and 8:18 is 8 min. The gap between 8:19 and 8:30 is 11 min. The gap between 8:31 and the next opening at 8:54 is 23 minutes. This is greater than 15 minutes. So, the first "ON" period ends 15 minutes after the 8:31 opening.
Light turns OFF at: 8:31 + 15 min = 8:46.
Light was ON from 8:00 to 8:46.
OFF Period 1: The light is off from 8:46 until the next opening at 8:54.
Duration OFF: 8:54 - 8:46 = 8 minutes.
ON Period 2: The light turns on at 8:54. The sequence is 8:54, 8:57, 9:05, 9:11. The gap between 9:11 and the next opening at 9:29 is 18 minutes (>15 min). So, this ON period ends 15 minutes after 9:11.
Light turns OFF at: 9:11 + 15 min = 9:26.
Light was ON from 8:54 to 9:26.
OFF Period 2: The light is off from 9:26 until the next opening at 9:29.
Duration OFF: 9:29 - 9:26 = 3 minutes.
ON Period 3: The light turns on at 9:29. The sequence is 9:29, 9:31. There are no more openings until 10:00. This ON period ends 15 minutes after the final opening at 9:31.
Light turns OFF at: 9:31 + 15 min = 9:46.
Light was ON from 9:29 to 9:46.
OFF Period 3: The light is off from 9:46 until the end of the period at 10:00.
Duration OFF: 10:00 - 9:46 = 14 minutes.
Step 3: Final Answer:
Total time the light was off = (OFF Period 1) + (OFF Period 2) + (OFF Period 3)
Total OFF time = 8 + 3 + 14 = 25 minutes. This corresponds to option (2).
