Answer to Question #99243 in Differential Equations for Leo

Question

Newtons laws of cooling proposes that the rate of change of temperature is proportional to the temperature difference to the ambient (room) temperature. And can be modelled using the equation: dT/dt = -k (T-Ta)

It can also be written as dT/T-Ta = -k dt

Where:
T = Temperature of material
Ta = Ambient (room) temperature
k = A cooling constant

a) integrate both sides of the equation and show that the temperature difference is given by:

(T-Ta) = CoE^-kt

(Co is a constant for this problem)

B) calculate Co if the initial temperature is 70 degrees C and Ta = 20 degrees C?

Answer 1

A) 

$\int \cfrac{dT}{T-T_a} = - \int k\,dt$

$ln\,|T - T_a| = -kt + ln\,C_0$

$T - T_a = e^{-kt + ln\,C_0}$

$T - T_a = e^{ln\,C_0}\,e^{-kt}$

$T - T_a = C_0\,e^{-kt}$

B)

$70 - 20 = C_0\,e^{-k*0} \\ \underline {C_0 = 50}$