Question

A body is kept in cooler surroundings. When the temperature of the body is ${70}^{0}C$, it cools at the rate of $3^{0}C$ per minute and at ${50}^{0}C$, it cools at the rate of ${1.5}^{0}C$ per minute. Find the temperature of the body when it cools at the rate of $1^{0}C$ per minute.

Answer 1

Rate of cooling $\frac{dT}{dt}=\frac{-k}{ms}(T-T_s)$

Or $3=\frac{-k}{ms}(70-T_s)$ .....(1)

Or $1.5=\frac{-k}{ms}(50-T_s)$ ......(2)

So, $\frac{3}{1.5}=\frac{70-T_s}{50-T_s}$ $⟹T_s={30}^{o}C$

When rate of cooling is $1^{o}C$ per minute

So, $1=\frac{-k}{ms}(T-30)$ .....(3)

Equation (3)/(1), $\frac{1}{3}=\frac{T-30}{70-30}$

$⟹T={43.33}^{o}C$