Question

In $5\text{minutes}$, a body cools from $75{}^{\circ }\text{C}$ to $65{}^{\circ }\text{C}$ at room temperature of $25{}^{\circ }\text{C}$. The temperature of the body at the end of next $5\text{minutes}$ is ${}^{\circ }\text{C}$

Answer 1

By newton’s law of cooling (with approximation),

$\frac{\Delta T}{\Delta t}=-k(T_{avg}-T_s)$

For $1^{st}5$ minutes,

$\frac{-10{}^{\circ }\text{C}}{5\text{min}}=-k(70{}^{\circ }\text{C}-25{}^{\circ }\text{C})$

$\Rightarrow k=\frac{2}{45}{\text{min}}^{-1}$

Let the temperature of the body at the end of next $5\text{minutes}$ be $T{}^{\circ }\text{C}$.

$\frac{T-65}{5\text{min}}=-k(\frac{T+65}{2}-25)=-\left(\frac{2}{45}\right)\left(\frac{T+15}{2}\right)$

$\Rightarrow 9(T-65)=-(T+15)$

$\Rightarrow 10T=570$

$\Rightarrow T=57{}^{\circ }\text{C}$

Correct answer: $57$