Newton's law of cooling states that the rate of loss of temperature of a cooling body is proportiona

Question

Newton's law of cooling states that the rate of loss of temperature of a cooling body is proportional to the difference in temperature between the body ($\theta{}^\circ$) and its surroundings ($\theta_s{}^\circ$). Express this law in the form of a differential equation, using the symbols $t$ for time and $\theta$ for the temperature of the body. Solve this equation for $\theta$ in terms of $t$, given that the room temperature is $20{}^\circ$C and that it takes a particular body $12$ minutes to cool from $100{}^\circ$C to $50{}^\circ$C. Hence find the time taken by the body to cool from $50{}^\circ$C to $25{}^\circ$C.

Accepted Answer

(a) $\frac{\mathrm d\theta}{\mathrm dt}=-k(\theta-\theta_s)$, $k>0$. (b) $\theta=20+80\left(\frac38\right)^{\frac t{12}}$. (c) $\frac{12\ln6}{\ln\frac83}\approx21.9$ minutes.

Word problem: modelling with differential equations

Word problem: modelling with differential equations

Solution:

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