Question

A body with an initial temperature ${\theta }_1$ is allowed to cool in a surrounding which is at a constant temperature of ${\theta }_0({\theta }_0<{\theta }_1)$. Assume that Newton's law of cooling is obeyed. The temperature of the body after time $t$ is best expressed by, Let $k=constant$.

• A. $({\theta }_1-{\theta }_0)e^{-kt}$
• B. $({\theta }_1-{\theta }_0)ln\left(kt\right)$
• C. ${\theta }_0+({\theta }_1-{\theta }_0)e^{-kt}$
• D. ${\theta }_1{e}^{-kt}={\theta }_0$

Answer 1

The correct option is C ${\theta }_0+({\theta }_1-{\theta }_0)e^{-kt}$

$\frac{d\theta }{dt}=-k(\theta -{\theta }_0)$
${\int }_{{\theta }_1}^{\theta }\frac{d\theta }{\theta -{\theta }_0}=-k{\int }_0^{t}dt=\theta ={\theta }_0+({\theta }_1-{\theta }_0)e^{-kt}$
Hence (C) is correct.