1. Suppose that a metal rod of length 40 cm is heated to a temperature of 80 degrees Celsius, and the ends are kept at 0 degrees. Assume that the diffusion coefficient a = 1 You will get the following set of equations: ди ²u at əx² u(x,0) = 80 u(0, t) = 0, u(40, t) = 0 a. Jse separation of variables to solve this equation. Show all your work. b. Write the solution as an infinite sum. c. Sketch a graph of the solutions at t=0, t=1, t=10, t=50. d. Which point/points on the metal rod are the hottest as time increases. e. Approximately how long will it take for the entire rod to cool below 5 degrees Celsius? You may use a single term expansion of the series in your approximation. f. Describe following limit and find the solution: lim u(x, t). t→∞

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