A patient receives a medication intravenously according to the following delivery schedule. For the first 60 minutes after the infusion begins, the rate of delivery is
D(t)=120e−0.02tmgmin−1,0≤t≤60.
Beginning at t=60 minutes, a constant maintenance rate of 40mgmin−1 is administered.
Let M(t) be the total amount of medication, in milligrams, that has entered the patient by time t minutes.
(a) Set up, but do not evaluate, a piecewise definite integral expression for M(t) valid for all t≥0.
(b) Evaluate the integral from part (a) to obtain a piecewise-defined formula for M(t).
(c) The treatment protocol calls for the patient to receive a total of 3,000mg. Determine the exact time (in minutes) at which this total amount is first reached. Clearly indicate whether the target is achieved during the initial delivery phase (0≤t≤60) or during the maintenance phase (t>60). Show all supporting calculations.
(d) Justify that the time found in part (c) is the only time at which the cumulative dose reaches 3,000mg.