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Unformatted text preview: AMATH 351 Assignment No. 1 Fall 2005 Warmup/Review/Introduction For review and not to be handed in: AM351 Course Notes, Problem Set No. 1, Questions R1,R2,R3 (attached to this assignment). The following problems are due Wednesday, September 21, 2005. 1. Suppose that a given radioactive element A decomposes into a second radioactive element B and that B in turn decomposes into a third element C that is not radioactive. Also let k 1 and k 2 denote the rate constants of these two decay processes. Let x > 0 be the amount of A present initially and the amounts of A and B present at a later time t be x ( t ) and y ( t ), respectively, with y (0) = 0. (a) Determine x ( t ) and y ( t ) for t ≥ 0. Consider the two cases (i) k 1 negationslash = k 2 and (ii) k 1 = k 2 . Sketch graphs of y ( t ) for both cases, noting any important qualitative behaviour, e.g. max- ima/minima, zeros, asymptotes. What is z ( t ), the concentration of C in each case? (Hint: You do not have to solve a DE for z ( t ).) (b) Show that for a fixed t ≥ 0, the solution y ( t ) of (i) becomes the solution...
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