Q.79 - Calculus

Question

Use the solution of the differential equation $\frac{dT}{dt} = k(A - T)$ for the Newton’s Law of Cooling and Heating model to prove that as t → ∞, the temperature T(t) of an object approaches the ambient temperature A of its environment. The proof requires that we assume that k is positive. Why does this make sense regardless of whether the model represents heating or cooling?

Step-by-Step Solution

Answer

$T\left(t\right)=A-(A-T_0)e^{-kt}$

Step 1. Given information

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

Step 2. Integrating both the sides

$\int \frac{dT}{(A-T)}=\int kdt$

$-\ln (A-T)=Kt+C1$

$\ln (A-T)=-Kt-C1$

Since,

$A-T=C{e}^{-kt}$

$T=A-C{e}^{-kt}$

Step 3. Using initial conditions

Using initial conditions, we get,

Substitute the value of C,