Answer: Suppose that as a body cools, the temperature of the surrounding medium
Chapter 2, Problem 35(choose chapter or problem)
Suppose that as a body cools, the temperature of the surrounding medium increases because it completely absorbs the heat being lost by the body. Let T(t) and \(T_{m}(t)\) be the temperatures
of the body and the medium at time t, respectively. If the initial temperature of the body is \(T_{1}\) and the initial temperature of the medium is \(T_{2}\), then it can be shown in this case that Newton’s law of cooling is \(d T / d t=k\left(T-T_{m}\right), k<0\), where \(T_{m}=T_{2}+B\left(T_{1}-T\right), B>0\) is a constant.
(a) The foregoing DE is autonomous. Use the phase portrait concept of Section 2.1 to determine the limiting value of the temperature T(t) as \(t \rightarrow \infty\). What is the limiting value
of \(T_{m}(t)\) as \(t \rightarrow \infty\)?
(b) Verify your answers in part (a) by actually solving the differential equation.
(c) Discuss a physical interpretation of your answers in part (a).
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