The temperature u(x,t) in a bar of silver of length L = 10cm, density p = 10.6-, thermal cal cal cm conductivity K 1.04 , and specific heater = 0.056 that is laterally insulated cm * deg * s gm * deg and whose ends are kept at 0 oC is governed by the heat equation d 8 2 u(x, t) B m(x, r), 0 < x < L, 0 < r dt dxwith boundary conditions m(0, r) = 0 and u{L,t) 0 and initial condition m(x,0) = /(x) = lOx x 2 . The solution to the problem is given by 00 , "(x, 0 = ^a exp ( =i ^ P 2 n 2 n 2 \ . fnn -t sin x where fi = ^ and the coefficients are from the Fouriersine series a i< s ' n {t'x ) f or f(x ) where a " = I Jo /Wsin {tx ) cIx - a. Find the first four nonzero terms ofthe Fourier sine series of f(x) = lOx x 2 . b. Compare f(x) to the first four nonzero terms of a\ sin (^i) + 02 sin (2^) + cit, sin (3^) + fla sin (4^) + for x = 3, 6. 9. c. Find the first four nonzero terms of u{x,t). d. Approximate m(9, 0.5), t((6, 0.75), m(3, 1) using part (c).

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