A certain hormone is produced by an endocrine gland, causing its concentration to increase at a constant rate of A mg/L/hr. The hormone is metabolized by the liver, with the rate of elimination being proportional to the concentration of the hormone. The constant of proportionality in this relationship is given by the (positive) constant k. The concentration of the hormone at time t is written as y(t). (a) Write down the right side of the differential equation satisfied by y. (Your answer should be in terms of y.) dy dt (b) This differential equation can be rewritten in the more familiar form = k(M - y) for a particular choice of M. Write down an expression for M in terms of A and k. M = (c) If the initial concentration is zero, the solution of the differential equation is y(t) = M(1 - ecki). In terms of A and k, what value does the hormone concentration approach in the long run? mg/L At time t = 0, a patient comes off a drug treatment that had blocked production of the hormone. (So, at that time, they do not have any of the hormone in their body.) Their doctor finds that the initial rate of increase of their hormone concentration is 1.1 mg/L/hr and that, in the long run, the concentration approaches 4 mg/L. (Give your answers correct to at least three decimal places.) (d) After how long would the patient's hormone concentration have reached half of its long-term level? hours (e) At what rate would the level of hormone have been increasing at that time? mg/L/hr