Technology Exercise
Therapeutic Level of a Drug 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
f
(
t
)
.
Write a differential equation that is satisfied by the amount
f
(
t
)
of the drug in the blood at time
t
(in hours).
Find
f
(
t
)
assuming that
f
(
0
)
=
0
(Give your answer in terms of
r
.)
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
,
...
,
2.
By looking at the graphs, pick the values of
r
that yield a therapeutic and nontoxic
2
-hour infusion.