10. (a) Let the ends of a copper rod 100 cm long be main- tained at 0°C. Suppose that the center of the rod is heated to 100°C by an external heat source and that this situation is maintained until a steady state results. Find this steady-state temperature distribution. (b) At a time t = 0 [after the steady state of part (a) has been reached], let the heat source be removed. At the same instant let the end x = 0 be placed in thermal contact with a reser- voir at 20°C, while the other end remains at 0°C. Find the temperature as a function of position and time. (c) Plot u versus x for several values of t. Also plot u versus t for several values of x. (d) What limiting value does the temperature at the center of the rod approach after a long time? How much time must elapse before the center of the rod cools to within 1° of its limiting value?