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 .
- The AC setting was changed (including turning it on/off, and/or setting different temperatures) only at the beginning of the hour or at minutes after the hour.
- The AC was used in POWER mode for longer duration than in REGULAR mode during this3-hour period.
Solution
Firstly, let's understand what the set says:
- There are two ways to operate the AC - regular and power.
- There is a set temperature of the AC. It is a temperature that the AC will automatically reach when it is off.
- Regular mode of the AC brings down the temperature inside the room (called inside temperature) at a constant rate to the set temperature in 1 hour.
- Power mode of the AC brings down the temperature inside the room (called inside temperature) at a constant rate to the set temperature in 30 minutes.
- When the AC is turned off, the temperature rises linearly, reaching the temperature at the time it was turned off in one hour.
- The temperature outside has been falling at a constant rate from 7 pm onward until 3 am on a particular night.
- The graph in the question describes the inside temperature of the room every 30 minutes starting from 23:00.
| Time | Temperature | AC - on / off |
|---|---|---|
| 23:00 | 38 | on |
| 23:30 | 32 | on |
| 00:00 | 26 | off |
| 00:30 | 31 | on |
| 01:00 | 26 | off |
| 01:30 | 30 | on |
| 02:00 | 28 |
This is the table we can directly make from the graph.
Now, we will first try to find out when the AC was on or off.
From 23:00 to 23:30, the temperature decreased, which means the AC was on.
From 23:30 to 00:00, the temperature decreased, which means the AC was on.
From 00:00 to 00:30, the temperature increased, which means the AC was off.
From 00:30 to 01:00, the temperature decreased, which means the AC was on.
From 01:00 to 01:30, the temperature increased, which means the AC was off.
From 01:30 to 02:00, the temperature decreased, which means the AC was on.
| Time | Temperature | AC - on / off |
|---|---|---|
| 23:00 | 38 | on |
| 23:30 | 32 | on |
| 00:00 | 26 | off |
| 00:30 | 31 | on |
| 01:00 | 26 | off |
| 01:30 | 30 | on |
| 02:00 | 28 |
From the given information,
We know the AC was started for the first time at 23:00.
This simply means that before the AC was turned on, the room temperature was the same as the outside temperature. Therefore, the temperature at 23:00 (which is 38 degrees) is the outside temperature at that time.
| Time | Temperature | AC - on / off | Outside temperature |
|---|---|---|---|
| 23:00 | 38 | on | 38 |
| 23:30 | 32 | on | |
| 00:00 | 26 | off | 36 |
| 00:30 | 31 | on | |
| 01:00 | 26 | off | |
| 01:30 | 30 | on | |
| 02:00 | 28 |
We know from the table that the AC was turned off at 00:00.
According to the question, when the AC is off, the room temperature must reach the outside temperature in 1 hour.
The temperature rises from 26 degrees to 31 degrees at 00:30 (an increase of 5 degrees in 30 minutes).
Since the temperature rises at a constant rate, this means there would be an increase of 10 degrees in 1 hour (5 degrees × 2 = 10 degrees).
Therefore, the outside temperature at 00:00 was 26 + 10 = 36 degrees.
| Time | Temperature | AC - on / off | Outside temperature |
|---|---|---|---|
| 23:00 | 38 | on | 38 |
| 23:30 | 32 | on | 37 |
| 00:00 | 26 | off | 36 |
| 00:30 | 31 | on | 35 |
| 01:00 | 26 | off | 34 |
| 01:30 | 30 | on | 33 |
| 02:00 | 28 | 32 |
We also know that the outside temperature has decreased at a constant rate.
We found that it was 38 degrees at 23:00 and 36 degrees at 00:00.
This means the temperature decreased by 2 degrees in 1 hour (60 minutes). Since the rate is constant, this equals 1 degree every 30 minutes.
Since the temperature decreases at a constant rate, at 23:30 it will be 37 degrees.
From this, we can now find the rate at which the outside temperature has decreased, which is 1 degree every 30 minutes.
| Time | Temperature | AC - On / Off | Outside temperature |
|---|---|---|---|
| 23:00 | 38 | ON | 38 |
| 23:30 | 32 | ON | 37 |
| 00:00 | 26 | OFF | 36 |
| 00:30 | 31 | ON | 35 |
| 01:00 | 26 | OFF | 34 |
| 01:30 | 30 | ON | 33 |
| 02:00 | 28 | 32 |
We know that the AC got turned ON after 11:01 PM at 12:30 AM and 1:30 AM.
Hence, the answer is 'Exactly 3'