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6. (18 points) It is well-known that the rate of change of the amount of a radioac- tive element at a given time is directly proportional to the amount present at that time. A radioactive substance A decays into some other radioactive substance B with decay constant a > 0. Moreover the substance B also decays with decay constant b>0. Let t denote the time and let A(t) and B(t) denote the amount of the elements A and B at time t respectively. Assume that initially at t = 0, there is no element B and the amount of element A is A₁. Write differential equations describing the process and solve them to obtain an expression for B(t). What is the amount of element B when t = 1? (Assume that a b).