Question

Newton's law of cooling states that the rate of change of temperature of an object in a surrounding medium is proportional to the difference of the temperature of the medium and the temperature of the object. Suppose a metal bar, initially at temperature 60 degrees Celsius, is placed in a room which is held at the constant temperature of 30 degrees Celsius. One minute later the bar has cooled to 30.54947 degrees.
Write the differential equation that models the temperature in the bar (in degrees Celsius) as a function of time (in minutes).

Hint: You will need to find the constant of proportionality. Start by calling the constant k and solving the initial value problem to obtain the temperature as a function of k and t. Then use the observed temperature after one minute to solve for k .

Answer 1

Answer:

Check the explanation

Step-by-step explanation:

The differential equation governing the given situation is,

$\frac{dT}{dt}=-k\left ( T-T_s \right )$

Integrating the above we get,

=>$\int \frac{\mathrm{d}T }{T-T_s}=\int -kdt$

=>$\ln(T-T_s)=-kt+C$

At t=0 T=T0, applying this condition in the above equation we get

=>$T(t)-T_s=(T(0)-T_s)e^{-kt}$

It is given that it takes 1 minute for the temperature to fall from 30 to 10.99574 with T(t)=10.99574, T(0)=30 and Ts=10

=>$10.99574-10=(30-10)e^{-kt}$

=>$0.99574=20e^{-kt}$

=>$e^{kt}=\frac{20}{0.99574}$

=>$e^{k\times 1}=\frac{20}{0.99574}$

=>$k=\ln\frac{20}{0.99574}=3$