A substance cools from `75^@C` to` 70^@C` in `T_1` minute, from `70^@C` to `65^@C` in `T_2` minute and from `65^@C to 60^@C` in `T_3` minute, then.

Options

A. `T_1 = T_2 = T_3`
B. `T_1ltT_2ltT_3`
C. `T_1gtT_2gtT_3`
D. `T_(1)gtT_(2)=T_(3)`

Explanation

To solve the problem, we need to analyze the cooling process of the substance according to Newton's Law of Cooling. Let's break it down step by step.

### Step 1: Understand the cooling process
The substance cools from:
- 75°C to 70°C in \( T_1 \) minutes
- 70°C to 65°C in \( T_2 \) minutes
- 65°C to 60°C in \( T_3 \) minutes

### Step 2: Apply Newton's Law of Cooling
According to Newton's Law of Cooling, the rate of change of temperature of an object is directly proportional to the difference between its temperature and the ambient temperature. Mathematically, this can be expressed as:

\[ \frac{dT}{dt} \propto (T - T_0) \]

where \( T \) is the temperature of the substance and \( T_0 \) is the surrounding temperature.

### Step 3: Identify average temperatures
For each cooling interval, we can calculate the average temperature:
- For \( T_1 \): Average temperature \( T_{avg1} = \frac{75 + 70}{2} = 72.5°C \)
- For \( T_2 \): Average temperature \( T_{avg2} = \frac{70 + 65}{2} = 67.5°C \)
- For \( T_3 \): Average temperature \( T_{avg3} = \frac{65 + 60}{2} = 62.5°C \)

### Step 4: Analyze the relationship between time and average temperature
From Newton's Law of Cooling, we know that the time taken for cooling is inversely proportional to the temperature difference:

\[ T \propto \frac{1}{(T_{avg} - T_0)} \]

Since \( T_0 \) (the surrounding temperature) is constant, we can say:

\[ T_1 \propto \frac{1}{(72.5 - T_0)}, \quad T_2 \propto \frac{1}{(67.5 - T_0)}, \quad T_3 \propto \frac{1}{(62.5 - T_0)} \]

### Step 5: Compare the average temperatures
Since \( T_{avg1} > T_{avg2} > T_{avg3} \), it follows that:

\[ \frac{1}{(72.5 - T_0)} < \frac{1}{(67.5 - T_0)} < \frac{1}{(62.5 - T_0)} \]

This implies:

\[ T_1 < T_2 < T_3 \]

### Conclusion
Thus, the relationship between the times taken for cooling is:

\[ T_1 < T_2 < T_3 \]

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

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