A body is cooled in 2 min n a room at temperature of `30^@C` from `75^@C` to `65^@C`. If the same body Is cooled from `55^@C` to `45^@C` in the same room, find the time taken (in minute).

Explanation

To solve the problem of how long it takes for a body to cool from 55°C to 45°C in a room at 30°C, we can use Newton's Law of Cooling. Here’s a step-by-step solution:

### Step 1: Understand Newton's Law of Cooling
Newton's Law of Cooling states that the rate of change of temperature of an object is proportional to the difference between its own temperature and the ambient temperature (the temperature of the surroundings).

### Step 2: Identify the Given Data
- Initial temperature (T1) = 75°C
- Final temperature (T2) = 65°C
- Ambient temperature (T_ambient) = 30°C
- Time taken for the first cooling (t1) = 2 minutes

### Step 3: Calculate the Temperature Differences
For the first cooling:
- Change in temperature (ΔT1) = T1 - T2 = 75°C - 65°C = 10°C
- Average temperature during cooling (T_avg1) = (T1 + T2) / 2 = (75°C + 65°C) / 2 = 70°C

For the second cooling:
- Initial temperature (T3) = 55°C
- Final temperature (T4) = 45°C
- Change in temperature (ΔT2) = T3 - T4 = 55°C - 45°C = 10°C
- Average temperature during cooling (T_avg2) = (T3 + T4) / 2 = (55°C + 45°C) / 2 = 50°C

### Step 4: Calculate the Temperature Differences with Ambient Temperature
For the first case:
- Temperature difference with ambient (D1) = T_avg1 - T_ambient = 70°C - 30°C = 40°C

For the second case:
- Temperature difference with ambient (D2) = T_avg2 - T_ambient = 50°C - 30°C = 20°C

### Step 5: Set Up the Proportionality
According to Newton's Law of Cooling, the time taken is inversely proportional to the temperature difference:
\[ t \propto \frac{\Delta T}{D} \]
Thus, we can write:
\[ \frac{t1}{t2} = \frac{D2}{D1} \]

### Step 6: Substitute the Values
We know:
- t1 = 2 minutes
- D1 = 40°C
- D2 = 20°C

Now substituting these values:
\[ \frac{2}{t2} = \frac{20}{40} \]
\[ \frac{2}{t2} = \frac{1}{2} \]

### Step 7: Solve for t2
Cross-multiplying gives:
\[ 2 \cdot 2 = 1 \cdot t2 \]
\[ t2 = 4 \text{ minutes} \]

### Final Answer
The time taken for the body to cool from 55°C to 45°C is **4 minutes**.

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