Chapter 7, Problem 7P

To determine

To calculate : the temperature of the room

Answer to Problem 7P

The temperature of the room is $T_{room}={20}^{\circ }C$

Explanation of Solution

Given information : The question is

A peach pie is taken out of the oven at $5:00pm$ . At that time it is piping hot, ${100}^{\circ }C$ at $5:10pm$ its temperature is ${80}^{\circ }C$ at $5:20pm$ it is ${65}^{\circ }C$

Calculation :

Using the Newton’s law of cooling

  $\begin{array}{l}\frac{dT}{dt}\propto -\left(T-T_{room}\right)\\ \frac{dT}{dt}=-k\left(T-T_{room}\right)\left(\text{substitute }T-T_{room}=u\text{ and }\frac{dT}{dt}=\frac{du}{dt}\right)\\ \frac{du}{dt}=-ku\\ u\left(t\right)=u\left(0\right)e^{-\raisebox{1ex}{$t$}\!\left/ \!\raisebox{-1ex}{$k$}\right.}\\ T-T_{room}=\left[T\left(0\right)-T_{room}\right]e^{-\raisebox{1ex}{$t$}\!\left/ \!\raisebox{-1ex}{$k$}\right.}\\ T\left(t\right)-T_{room}=\left[100-T_{room}\right]e^{-\raisebox{1ex}{$t$}\!\left/ \!\raisebox{-1ex}{$k$}\right.}\left(\text{at }5:10pm\text{ temperature is }{80}^{\circ }C\text{ T}\left(10\right)\text{=80 }\right)\\ 80-T_{room}=\left[100-T_{room}\right]e^{-\raisebox{1ex}{$10$}\!\left/ \!\raisebox{-1ex}{$k$}\right.}\mathrm{......}\left(1\right)\\ 65-T_{room}=\left[100-T_{room}\right]e^{-\raisebox{1ex}{$20$}\!\left/ \!\raisebox{-1ex}{$k$}\right.}\\ 65-T_{room}=\left[100-T_{room}\right]{\left(e^{-\raisebox{1ex}{$10$}\!\left/ \!\raisebox{-1ex}{$k$}\right.}\right)}^2\end{array}$

Use equation 1 $\frac{80-T_{room}}{100-T_{room}}=e^{-\raisebox{1ex}{$10$}\!\left/ \!\raisebox{-1ex}{$k$}\right.}$

  $\begin{array}{l}65-T_{room}=\left[100-T_{room}\right]\left[\frac{80-T_{room}}{100-T_{room}}\right]\\ 65-T_{room}=\frac{{\left(80-T_{room}\right)}^2}{100-T_{room}}\end{array}$

Replace $T_{room}$ with $x$

  $\begin{array}{l}65-x=\frac{{\left(80-x\right)}^2}{100-x}\\ \left(65-x\right)\left(100-x\right)={\left(80-x\right)}^2\\ 65\left(100-x\right)-x\left(100-x\right)={80}^2-2\left(x\right)\left(80\right)+x^2\\ 6500-65x-100x+x^2=6400-160x+x^2\\ 100=5x\\ x=20\\ T_{room}={20}^{\circ }C\end{array}$