A bucket full of hot water cools from `75^(@)C` to `70^(@)C` in time `T_(1)`, from `70^(@)C` to `65^(@)C` in time `T_(2)` and from `65^(@)C` to `60^(@)C` in time `T_(3)`, then

Options

A. `T_1= T_2 = T_3`
B. `T_1 gt T_2 gt T_3`
C. `T_1 lt T_2 lt T_3`
D. `T_1 gt T_2 lt T_3`

Answer

To solve the problem of the cooling of a bucket of hot water, we can use 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.

### Step-by-Step Solution:

1. **Identify the Temperature Changes**:
- The water cools from 75°C to 70°C in time \( T_1 \).
- It cools from 70°C to 65°C in time \( T_2 \).
- It cools from 65°C to 60°C in time \( T_3 \).

2. **Calculate the Temperature Differences**:
- The temperature difference for each interval is:
- From 75°C to 70°C: \( \Delta T_1 = 75 - 70 = 5°C \)
- From 70°C to 65°C: \( \Delta T_2 = 70 - 65 = 5°C \)
- From 65°C to 60°C: \( \Delta T_3 = 65 - 60 = 5°C \)

3. **Apply Newton's Law of Cooling**:
- According to Newton's Law of Cooling, the rate of cooling is proportional to the temperature difference between the object and its surroundings:
\[ \frac{dT}{dt} \propto (T - T_s) \]
- Here, \( T \) is the temperature of the water and \( T_s \) is the surrounding temperature.

4. **Determine Average Temperatures**:
- For each cooling interval, we can calculate the average temperature:
- \( \theta_1 = \frac{75 + 70}{2} = 72.5°C \)
- \( \theta_2 = \frac{70 + 65}{2} = 67.5°C \)
- \( \theta_3 = \frac{65 + 60}{2} = 62.5°C \)

5. **Analyze the Relationship**:
- The temperature difference between the water and the surrounding temperature decreases as the water cools.
- Since the temperature differences are constant (5°C), but the average temperatures are decreasing, we can infer that the time taken for cooling is inversely proportional to the temperature difference:
\[ T_1 > T_2 > T_3 \]
- This means that the time taken to cool from 75°C to 70°C (i.e., \( T_1 \)) is less than the time taken to cool from 70°C to 65°C (i.e., \( T_2 \)), which is less than the time taken to cool from 65°C to 60°C (i.e., \( T_3 \)).

6. **Conclusion**:
- Therefore, the relationship between the cooling times is:
\[ T_1 < T_2 < T_3 \]

### Final Answer:
The correct relationship is \( T_1 < T_2 < T_3 \).