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

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Answer

To solve the problem of how the temperature \( T \) of a piece of metal changes over time \( t \) as it cools in a room at temperature \( \theta_0 \), we can apply Newton's Law of Cooling. Here is a step-by-step solution:

### Step 1: Understand Newton's Law of Cooling
According to Newton's Law of Cooling, the rate of change of temperature of an object is proportional to the difference between its temperature and the ambient temperature. Mathematically, this can be expressed as:
\[ \frac{dT}{dt} = -k(T - \theta_0) \]
where:
- \( T \) is the temperature of the metal,
- \( \theta_0 \) is the ambient temperature,
- \( k \) is a positive constant.

### Step 2: Rearranging the Equation
We can rearrange the equation to isolate the variables:
\[ \frac{dT}{T - \theta_0} = -k \, dt \]

### Step 3: Integrate Both Sides
Next, we integrate both sides. The left side will be integrated with respect to \( T \) and the right side with respect to \( t \):
\[ \int \frac{dT}{T - \theta_0} = -k \int dt \]
This gives us:
\[ \ln |T - \theta_0| = -kt + C \]
where \( C \) is the constant of integration.

### Step 4: Solve for \( T \)
To solve for \( T \), we exponentiate both sides:
\[ |T - \theta_0| = e^{-kt + C} = e^C e^{-kt} \]
Let \( A = e^C \), then we can write:
\[ T - \theta_0 = A e^{-kt} \]
Thus, we have:
\[ T = \theta_0 + A e^{-kt} \]

### Step 5: Determine the Constant \( A \)
At \( t = 0 \), the initial temperature \( T(0) = \theta \):
\[ \theta = \theta_0 + A \]
This implies:
\[ A = \theta - \theta_0 \]
Substituting \( A \) back into the equation gives:
\[ T = \theta_0 + (\theta - \theta_0)e^{-kt} \]

### Step 6: Analyze the Behavior of the Function
As \( t \) approaches infinity, the term \( e^{-kt} \) approaches 0, leading to:
\[ T \to \theta_0 \]
At \( t = 0 \), \( T = \theta \). The graph of \( T \) versus \( t \) will show an exponential decay towards \( \theta_0 \).

### Conclusion
The graph of the temperature \( T \) of the metal versus time \( t \) will be an exponential decay curve approaching the ambient temperature \( \theta_0 \).