1. (a) (c) 2. (15 pts) Two identical copper rods, each 20 cm long have a thermal diffusivity of 1.17 cm/sec. They are originally at a constant temperature of 0 C. and 50 C., respectively. At time t = 0, the rods are placed end-to-end so that heat can flow freely between them. The outer ends of the new 40-cm rod are held at a temperature of 0 C. Write down a boundary-value problem for the temperature u(x, t) of the 40-cm rod at any time and location. Find the series solution of the problem in part (a) by the methods we developed in class. Use the first term of the series solution to estimate the time required for the center of the 40- cm rod (at x = 20) to reach 5 C. (15 pts) Same as problem 1, accept the left end of the combined rod (at x = 0) is held at zero temperature and the right end (at x = 40) is insulated. 2860