A certain drug is administered intravenously to a patient at the continuous rate of r milligrams per hour. The patient’s body removes the drug from the bloodstream at a rate proportional to the amount of the drug in the blood, with constant of proportionality k = .5.

(a) Write a differential equation that is satisfied by the amount f(t) of the drug in the blood at time t (in hours).

(b) Find f (t) assuming that f (0) = 0. (Give your answer in terms of r.)

(c) In a therapeutic 2-hour infusion, the amount of drug in the body should reach 1 milligram within 1 hour of administration and stay above this level for another hour. However, to avoid toxicity, the amount of drug in the body should not exceed 2 milligrams at any time. Plot the graph of f (t) on the interval 1 ≤ t ≤ 2, as r varies between 1 and 2 by increments of .1. That is, plot f (t) for r = 1, 1.1, 1.2, 1.3,c, 2. By looking at the graphs, pick the values of r that yield a therapeutic and nontoxic 2-hour infusion.