A vessel full of hot water is kept in a room and it cools from `80^(@)C` to `75^(@)C` in `T_(1)` minutes, from `75^(@)C` to `70^(@)C` in `T_(2)` minutes and from `70^(@)C` to `65^(@)C` in `T_(3)` minutes Then .

Options

A. `t_(1) = t_(2) = t_(3)`
B. `t_(1) lt t_(2) = t_(3)`
C. `t_(1) lt t_(2) lt t_(3)`
D. `t_(1) gt t_(2) gt t_(3)`

Answer

To solve the problem, we will apply Newton's Law of Cooling, which states that the rate of change of temperature of an object is proportional to the difference between its temperature and the ambient temperature (surrounding temperature).

### Step-by-Step Solution:

1. **Understanding the Cooling Process**:
We have three cooling intervals:
- From 80°C to 75°C in \(T_1\) minutes.
- From 75°C to 70°C in \(T_2\) minutes.
- From 70°C to 65°C in \(T_3\) minutes.

2. **Applying Newton's Law of Cooling**:
According to Newton's Law of Cooling:
\[ \frac{d\theta}{dt} = -k(\theta - \theta_s) \]
where:
- \(\theta\) is the temperature of the object (water in this case),
- \(\theta_s\) is the surrounding temperature,
- \(k\) is a constant.

3. **Setting Up the Equations**:
For each cooling interval, we can express the time taken as:
- For the first interval (80°C to 75°C):
\[ T_1 \propto \frac{1}{\theta_{avg1} - \theta_s} \]
where \(\theta_{avg1} = \frac{80 + 75}{2} = 77.5°C\).

- For the second interval (75°C to 70°C):
\[ T_2 \propto \frac{1}{\theta_{avg2} - \theta_s} \]
where \(\theta_{avg2} = \frac{75 + 70}{2} = 72.5°C\).

- For the third interval (70°C to 65°C):
\[ T_3 \propto \frac{1}{\theta_{avg3} - \theta_s} \]
where \(\theta_{avg3} = \frac{70 + 65}{2} = 67.5°C\).

4. **Comparing the Average Temperatures**:
Since the surrounding temperature \(\theta_s\) remains constant, we can compare the average temperatures:
- \(\theta_{avg1} = 77.5°C\)
- \(\theta_{avg2} = 72.5°C\)
- \(\theta_{avg3} = 67.5°C\)

As the average temperature decreases, the time taken for cooling increases.

5. **Establishing the Relationship**:
From the proportionality established in step 3:
- Since \(\theta_{avg1} > \theta_{avg2} > \theta_{avg3}\), we have:
\[ T_1 < T_2 < T_3 \]

### Final Conclusion:
Thus, the relationship between the times taken for cooling is:
\[ T_3 > T_2 > T_1 \]