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 Tm(t) be the temperatures of the body and the medium at time t, respectively.

If the initial temperature of the body is T₁ and the initial temperature of the medium is T₂, then it can be shown in this case that Newton’s law of cooling is dT/dt = k(T - T_m), k < 0, where T_m = T₂ + B(T₁ - T), 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 → ∞. What is the limiting value of T_m(t) as t → ∞?

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

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