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Expert-verified Found in: Page 573 ### Calculus

Book edition 1st
Author(s) Peter Kohn, Laura Taalman
Pages 1155 pages
ISBN 9781429241861 # Use the solution of the differential equation $\frac{dT}{dt}=k\left(A-T\right)$ 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?

$T\left(t\right)=A-\left(A-{T}_{0}\right){e}^{-kt}$

See the step by step solution

## Step 1. Given information

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

## Step 2. Integrating both the sides

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

$-\mathrm{ln}\left(A-T\right)=Kt+C1$

$\mathrm{ln}\left(A-T\right)=-Kt-C1$

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, ### Want to see more solutions like these? 