A glucose solution is administered intravenously into the bloodstream at a constant rate r. As the glucose is added, it is converted into other substances and removed from the bloodstream at a rate that is proportional to the concentration at that time. Thus, a model for the concentration C = C (t) of the glucose solution in the bloodstream is dC/dt = r – kc where is a positive constant.
(a) Suppose that the concentration at time t = 0 is C0. Determine the concentration at any time by solving the differential equation.
(b) Assuming that C0 < r/k, find lim t→∞ C (t) and interpret your answer.