Solved: PROBLEM 7RPSuppose that as a body cools, the

Chapter , Problem 7RP

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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 \(t \longrightarrow \infty\). What is the limiting value of \(T_{m}(t) \text { as } t \rightarrow \infty\)?

(b) Verify your answers in part (a) by actually solving the differential equation.

(c) Discuss a physical interpretation of your answers in part (a).

Text Transcription:

T_m(t)

T_1

T_2

dT/dt  k(T - T_m)

T_m = T_2 + B(T_1 - T)

t longrightarrow infty

T_m(t)

t rightarrow infty