If a piece of metal is heated to temperature `theta` and the allowed to cool in a room which is at temperature `theta_0`, the graph between the temperature T of the metal and time t will be closet to

Options

A.
B.
C.
D.

Explanation

To solve the problem, we need to analyze the cooling of a heated piece of metal according to Newton's Law of Cooling. Let's break down the steps:

### Step-by-Step Solution:

1. **Understanding the Problem**:
- We have a piece of metal heated to a temperature \( \theta \).
- It is allowed to cool in a room at a constant temperature \( \theta_0 \).
- We need to determine the relationship between the temperature \( T \) of the metal and time \( t \).

2. **Applying Newton's Law of Cooling**:
- According to Newton's Law of Cooling, the rate of change of temperature of the object is proportional to the difference between its temperature and the ambient temperature.
- Mathematically, this can be expressed as:
\[ \frac{d\theta}{dt} = -k(\theta - \theta_0) \]
where \( k \) is a positive constant.

3. **Rearranging the Equation**:
- We can rearrange the equation to isolate the temperature terms:
\[ \frac{d\theta}{\theta - \theta_0} = -k \, dt \]

4. **Integrating Both Sides**:
- We integrate both sides:
\[ \int \frac{d\theta}{\theta - \theta_0} = -k \int dt \]
- This gives:
\[ \ln|\theta - \theta_0| = -kt + C \]
where \( C \) is the integration constant.

5. **Exponentiating to Solve for \( \theta \)**:
- To eliminate the logarithm, we exponentiate both sides:
\[ |\theta - \theta_0| = e^{-kt + C} = e^C e^{-kt} \]
- Let \( A = e^C \), we can write:
\[ \theta - \theta_0 = A e^{-kt} \]

6. **Expressing \( \theta \)**:
- Thus, we can express the temperature \( \theta \) as:
\[ \theta = \theta_0 + A e^{-kt} \]
- Here, \( A \) is determined by the initial condition (the initial temperature of the metal).

7. **Identifying the Graph**:
- As time \( t \) increases, the term \( A e^{-kt} \) decreases exponentially, meaning that the temperature \( \theta \) approaches \( \theta_0 \) asymptotically.
- The graph of \( \theta \) versus \( t \) will show an exponential decay towards \( \theta_0 \).

8. **Conclusion**:
- The graph will be a curve that starts at \( \theta \) and approaches \( \theta_0 \) over time, resembling an exponential decay function.
- Therefore, the correct answer is option **C**, which represents an exponential decrease.