(19-67) A cylindrical pipe has inner radius R_1 and outer radius R_2. The interior of the pipe carries hot water at temperature T_1. The temperature outside is T_2( smaller than T_1). (a) Show that the rate of heat loss for a length L of pipe is dQ/dt=2πk(T_1−T_2)L/ln(R_2/R_1), where k is the thermal conductivity of the pipe, (b) Suppose the pipe is steel with R_1=3.3cm, R_2=4.0cm, and T_2=18℃. If the pipe holds still water at T_1=71℃ , what will be the initial rate of change of its temperature? (c) Suppose water at 71 °C enters the pipe and moves at a speed of 8.0 cm/s. What will be its temperature drop per centimeter of travel?

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