A hospital patient receives intravenously 6 units of a certain drug per hour. The drug is eliminated exponentially, so that the fraction that remains in the patients body for t hours is $f ( t ) = e ^ { - t / 11 }$ . If the treatment is continued indefinitely, approximately how many units of the drug will be in the patient's body in the long run? Round to two decimal places, if necessary.
A) 17 units
B) 60 units
C) 66 units
D) 0.55 units

Question

Quizplus · Accepted Answer

The long-term amount of the drug in the patient's body can be found by calculating the steady-state level, which is the rate of drug administration divided by the rate of elimination. The rate of administration is 6 units per hour, and the rate of elimination is given by the half-life formula, which in this case is $1/11$ per hour (from the exponential decay function $e^{-t/11}$). The steady-state level is thus $6 / (1/11) = 6 \times 11 = 66$ units.

A Hospital Patient Receives Intravenously 6 Units of a Certain f(t)=e−t/11f ( t ) = e ^ { - t / 11 }f(t)=e−t/11

A Hospital Patient Receives Intravenously 6 Units of a Certain $f ( t ) = e ^ { - t / 11 }$