An intravenous infusion delivers an antibiotic to a patient at a constant rate of 40…
MATHEMATICS · Multi-part · AP · Calculus AB · Differential Equations · Differential Equations
Problem
An intravenous infusion delivers an antibiotic to a patient at a constant rate of 40 mg/min beginning at time t=0 minutes. The infusion is terminated after 30 minutes. Throughout and after the infusion, the drug is removed from the bloodstream by metabolism at a rate proportional to the amount of drug present, with proportionality constant 0.05 min−1. Let A(t) denote the amount of antibiotic (in milligrams) in the bloodstream at time t minutes.
(a) Write a differential equation satisfied by A(t) for the interval 0≤t≤30 and state the initial condition.
(b) Solve the differential equation from part (a) to obtain A(t) for 0≤t≤30.
(c) Immediately after the infusion stops (t>30), write a new differential equation governing A(t) for t≥30 and solve it subject to the appropriate continuity condition.
(d) Determine the maximum amount of antibiotic present in the bloodstream and the time at which it occurs. Show justification for both results.