Question

A body at an unknown temperature is placed in a room which is held at a constant temperature of ${30}^{0}F$. If after $10$ minutes the temperature of the body is $0^{0}F$ and after $20$ minutes the temperature of the body is ${15}^{0}F$, find the unknown initial temperature.

• A. $-50$
• B. $-20$
• C. $-40$
• D. $-30$

Answer 1

The correct option is D $-30$

We know from NEWTONS LAW OF COOLING that,

Rate of cooling of the body is proportional to difference between the body's temperature and the surrounding temperature.

$⟹-\frac{dT}{dt}=k(T-30)$

where $T$ is the body's temperature and $k$ is the proportionality constant,

separating the variables and integrating gives,

$\ln (T-30)=-kt+c$ where $c$ is the integration constant.

Given at $t=10min$, $T=0^{0}F$,

Substituting in the equation gives,

$\ln \left(\frac{30-T}{30}\right)=k(10-T)$

Also given that at $t=20min$,

Temperature of the body is ${15}^{0}F$

Substituting in the equation gives the value of proportionality constant $k$,

$\therefore k=\frac{\ln 2}{10}$

$⟹\ln \left(\frac{30-T}{T}\right)=\frac{\ln 2}{10}(10-t)$

To find the initial temperature we substitute $t=0$ which gives,

$T_{initial}=-{30}^{0}F$