Chapter 3.3, Problem 22E

Newton’s Law of Cooling/Warming As shown in Figure 3.3.12, a small metal bar is placed inside container A, and container A then is placed within a much larger container B. As the metal bar cools, the ambient temperature T_A(t) of the medium within container A changes according to Newton’s law of cooling. As container A cools, the temperature of the medium inside container B does not change significantly and can be considered to be a constant T_B. Construct a mathematical model for the temperatures T(t) and T_A(t), where T(t) is the temperature of the metal bar inside container A. As in Problems 1, 5, and 20, this model can be solved by using prior knowledge. Find a solution of the system subject to the initial conditions T(0) = T₀, T_A(0) = T₁.

FIGURE 3.3.12 Container within a container in Problem 22