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 of determining the correct graph between the temperature of a hot body kept in cooler surroundings and time, we will apply Newton's Law of Cooling. Here’s a step-by-step breakdown of the 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 temperature of the surroundings,
- \( k \) is a positive constant,
- \( t \) is time.

### Step 2: Set Up the Initial Conditions
Let:
- \( T_0 \) be the initial temperature of the hot body,
- \( T_s \) be the constant temperature of the surroundings.

Initially, at \( t = 0 \):
\[ T(0) = T_0 \]

### Step 3: Solve the Differential Equation
To find the temperature of the body as a function of time, we can integrate the differential equation. The solution to the equation is given by:

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

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

2. As \( t \to \infty \):
\[ T(t) \to T_s + (T_0 - T_s)e^{-\infty} = T_s + 0 = T_s \]
This indicates that the temperature of the body approaches the temperature of the surroundings as time progresses.

### Step 5: Graph the Function
The function \( T(t) = T_s + (T_0 - T_s)e^{-kt} \) is an exponential decay function. The graph will start at \( T_0 \) when \( t = 0 \) and will asymptotically approach \( T_s \) as \( t \) increases. The curve will be decreasing and will never actually reach \( T_s \), but will get infinitely close to it.

### Conclusion
The correct graph that represents the temperature of a hot body cooling in cooler surroundings over time is an exponential decay curve starting from \( T_0 \) and approaching \( T_s \).