Question 8

If a piece of metal is heated to temperature $$\theta$$ and then 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 closest to :

$$\frac{dT}{dt} = -k(T - \theta_0)$$

$$\int \frac{dT}{T - \theta_0} = -k \int dt$$

$$\ln(T - \theta_0) = -kt + C$$

At $$t = 0$$, the initial temperature $$T = \theta$$. Thus, the constant $$C = \ln(\theta - \theta_0)$$.

The temperature $$T$$ at any time $$t$$ is given by:

$$T - \theta_0 = (\theta - \theta_0)e^{-kt}$$

$$T = \theta_0 + (\theta - \theta_0)e^{-kt}$$

Graph (A) correctly shows this.

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