Question

(Newton's law of cooling/warming) Use 'separation of variables' to solve the IVP: 3+2-5 pts = k(90- T), k>0 with IC: T(0) 0. Estimate the constant k when T(1) 45 dt

Answer 1

Step-by-step explanation:

The Newton's law differential equation is:

$\frac{dT(t)}{dt} = k(90-T(t))$

We solve by variable separation

$\frac{dT(t)}{(90-T(t))} = kdt$

integrating both sides between time 0 to time t, T(0) = 0 to T(t).

$\int\limits^T_0 {\frac{dT}{90-T} } \ = \int\limits^t_0 {k} \, dt$

$-(ln(90-T)-ln(90-0)) = kt-k*0$

applying logarithm properties

$-ln(\frac{90-T}{90}) = kt$ (*)

applying exponential function to both sides

$(\frac{90-T}{90}) = e^{-kt}$

$T= 90-90e^{-kt}$

Now replacing the condition T(1) = 45 in (*)

$-(ln(\frac{90-45}{90}) = k*1$

$k = 0.6931$