The correct graph between the temperature of a hot body kept in cooler surrounding and time is (Assume newton's law of cooling)

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Answer

To solve the problem regarding the correct graph between the temperature of a hot body kept in cooler surroundings over time, we will 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. Mathematically, it can be expressed as:

\[ \frac{dT}{dt} = -k(T - T_s) \]

where:
- \( T \) is the temperature of the hot body,
- \( T_s \) is the surrounding temperature,
- \( k \) is a positive constant,
- \( t \) is time.

### Step 2: Set Initial Conditions
Let’s assume:
- The initial temperature of the hot body is \( T_0 \) (where \( T_0 > T_s \)).
- At time \( t = 0 \), the temperature of the body is \( T_0 \).

### Step 3: Solve the Differential Equation
To find the temperature of the body at any time \( t \), we can integrate the equation. The solution to the differential equation is given by:

\[ T(t) = T_s + (T_0 - T_s)e^{-kt} \]

### Step 4: Analyze the Temperature Over Time
1. At \( t = 0 \):
\[ T(0) = T_s + (T_0 - T_s)e^{0} = T_0 \]
This confirms that the initial temperature is \( T_0 \).

2. As \( t \to \infty \):
\[ T(t) \to T_s + (T_0 - T_s)e^{-\infty} = T_s \]
This indicates that the temperature of the body approaches the surrounding temperature \( T_s \) as time progresses.

### Step 5: Determine the Graph Shape
The equation \( T(t) = T_s + (T_0 - T_s)e^{-kt} \) shows that the temperature decreases exponentially over time. The graph of this function will start at \( T_0 \) and asymptotically approach \( T_s \) as \( t \) increases.

### Conclusion
The correct graph representing the temperature of a hot body cooling in a cooler surrounding over time will be an exponential decay curve starting from \( T_0 \) and approaching \( T_s \).