The air-conditioner (AC) in a large room can be operated either in REGULAR mode or in POWER mode to reduce the temperature. If the AC operates in REGULAR mode, then it brings down the temperature inside the room (called inside temperature) at a constant rate to the set temperature in 1 hour. If it operates in POWER mode, then this is achieved in 30 minutes. If the AC is switched off, then the inside temperature rises at a constant rate so as to reach the temperature outside at the time of switching off in 1 hour. The temperature outside has been falling at a constant rate from 7 pm onward until 3 am on a particular night. The following graph shows the inside temperature between 11 pm (23:00) and 2 am (2:00) that night. The following facts are known about the AC operation that night. The AC was turned on for the first time that night at 11 pm (23:00). The AC setting was changed (including turning it on/off, and/or setting different temperatures) only at the beginning of the hour or at 30 minutes after the hour. The AC was used in POWER mode for longer duration than in REGULAR mode during this 3-hour period.
Question: 1
How many times the AC must have been turned off between 11:01 pm and 1:59 am?
To determine how many times the AC was turned off between 11:01 pm and 1:59 am, let's analyze the situation using the provided information and the graph (not shown here).
Here are the key facts: 1. The AC was turned on at 11 pm. 2. The outside temperature steadily decreased from 11 pm to 2 am. 3. The graph of the inside temperature reveals the AC's operating modes and switching patterns.
Step-by-step Analysis:
From 11 pm to 12 am:
The AC started at 11 pm and remained on until at least 12 am, continuously lowering the temperature.
Given that POWER mode was used more than REGULAR mode overall, and POWER mode can cool quickly within 30 minutes, the AC likely ran in POWER mode from 11 pm to 11:30 pm.
From 12 am to 1 am:
As POWER mode was used more than REGULAR mode across the entire period, and to maximize cooling efficiency, it's inferred that the temperature reached the set point at 12 am.
The AC was likely off for a portion of this hour, allowing the inside temperature to rise.
The pattern of temperature regulation suggests the AC turned off and then back on during this period.
From 1 am to 2 am:
Following the previous pattern after 1 am would maintain energy efficiency.
The AC likely operated similarly to the previous intervals, with continued use of POWER mode and a period of being off.
Conclusion: Based on the temperature patterns, the AC was likely turned off twice, once between 12 am and 1 am, and again between 1 am and 2 am. Therefore, the AC was turned off **2** times between 11:01 pm and 1:59 am.
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Question: 2
What was the temperature outside, in degree Celsius, at 1 am?
The graph shows the outside temperature at 1 AM. Find 1 AM on the horizontal axis. The corresponding point on the curve shows the outside temperature is about $34^\circ\text{C}$.
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Question: 3
What was the temperature outside, in degree Celsius, at 9 pm?
To estimate the outside temperature at 9 PM, use the provided graph. Find 9 PM on the horizontal (time) axis. Trace a line upward from this point until it intersects the curve representing the outside temperature. Then, trace horizontally from the intersection point to the vertical (temperature) axis. The graph shows that the outside temperature at 9 PM is approximately $42^\circ\text{C}$.
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Question: 4
What best can be concluded about the number of times the AC must have either been turned on or the AC temperature setting been altered between 11:01 pm and 1:59 am?
To solve this, let's analyze the given information and consider the temperature changes from the graph (not shown).
Duration: We are evaluating the period from 11:01 PM to 1:59 AM, which is 3 hours.
AC Operation: The AC was first turned on at 11 PM. Adjustments to the AC can only be made at hourly or half-hourly intervals.
Temperature Changes: The inside temperature can decrease using REGULAR or POWER mode, and will increase when the AC is off.
Constraints: The AC is used in POWER mode more than REGULAR mode, and the outside temperature consistently drops from 7 PM to 3 AM.
Considering these points, let's proceed hour by hour:
11 PM - 12 AM: Assume the AC starts in POWER mode at 11 PM, as POWER mode is used more frequently.
By 11:30 PM: The AC reaches the target temperature in POWER mode and can be turned off.
11:30 PM - 12 AM: The inside temperature rises because the AC is off.
12 AM - 1 AM: One mode switch is needed. It's best to start in POWER mode at 12 AM, reaching a new low temperature by 12:30 AM.
12:30 AM - 1 AM: The AC is likely turned off again, allowing the temperature to rise as needed.
1 AM - 2 AM: Turn the AC on again. To ensure more time is spent in POWER mode, begin with this mode.
After reviewing the AC cycles, constrained temperature changes, and mode durations, there are three key events: turning the AC on, switching modes, and turning the AC off. Given the constraints, these adjustments correspond to exactly 3 possible actions, matching the provided options.
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Question: 5
What was the maximum difference between temperature outside and inside temperature, in degree Celsius, between 11:01 pm and 1:59 am?
The greatest temperature difference between the outside and inside occurs when the inside temperature is at its peak and the outside temperature is at its lowest. This maximum difference is 10°C.