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

Explanation:

The Newton's law differential equation is:

`(dT(t))/(dt) = k(90-T(t))`

We solve by variable separation

`(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 {(dT)/(90-T) } \ = \int\limits^t_0 {k} \, dt`

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

applying logarithm properties

`-ln((90-T)/(90)) = kt` (*)

applying exponential function to both sides

`((90-T)/(90)) = e^(-kt)`

`T= 90-90e^(-kt)`

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

`-(ln((90-45)/(90)) = k*1`

`k = 0.6931`