A pharmaceutical company models the amount (in milligrams) of a drug in a patient’s…

MATHEMATICS · Multi-part · AP · Calculus BC · Differential Equations · Differential Equations

Problem

A pharmaceutical company models the amount

Q(t)

(in milligrams) of a drug in a patient’s bloodstream with the differential equation

\frac{dQ}{dt} = 0.4\,Q - 0.0008\,Q^2

, where

t

is measured in minutes. (a) Express the differential equation in logistic form and identify the carrying capacity. (b) Determine the time rate of change of

Q

when

Q = 300\,\text{mg}

. (c) The therapeutic window for the drug lies between

200\,\text{mg}

and

600\,\text{mg}

. For which values of

Q

in this range is the concentration increasing? Justify your answer mathematically. (d) Suppose the drug is effective only while

Q

is rising. Starting from an initial dose of

100\,\text{mg}

, determine whether the effectiveness is increasing or decreasing after

15\,\text{min}

if the concentration at that time has reached

240\,\text{mg}

. Explain your reasoning using the model parameters.

Related theory

Browse the problem library