MATHEMATICS · Multi-part · AP · Calculus AB · Differential Equations · Differential Equations
Problem
Immediately after an intravenous injection, the concentration C(t) of a certain medication in a patient’s bloodstream is 12 mg/L. For this drug, the rate of change of concentration is proportional to the difference between the current concentration and 0 mg/L (complete elimination). One hour later, the concentration has decreased to 9 mg/L. Assume the proportionality constant remains unchanged for the relevant time period.
(1) Formulate a differential equation modeling C(t), clearly defining any parameters you introduce.
(2) Solve the differential equation to obtain C(t) and determine the exact value of the proportionality constant using the information given.
(3) Compute the half-life of the drug (the time required for C(t) to decline to 6 mg/L).
(4) Determine the time at which the concentration first falls below 2 mg/L, and comment on the clinical relevance of this result.