A hospital patient receives 4 units of a certain drug per hour intravenously. The drug is eliminated exponentially, so that the fraction that remains in the patients body for t hours is $f ( t ) = e ^ { - t / 12 }$ . If treatment is continued indefinitely, approximately how many units of the drug will be in the patient's body in the long run?
A) 44 units
B) 48 units
C) 52 units
D) 56 units

Question

Quizplus · Accepted Answer

The steady state occurs when the rate of drug input equals the rate of drug elimination. The rate of drug input is 4 units per hour. The rate of elimination can be found by considering the fraction eliminated per hour, which is given by the derivative of $f(t) = e^{-t/12}$. The derivative $f'(t) = -\frac{1}{12}e^{-t/12}$ represents the rate of change of the drug concentration. At steady state, the amount of drug in the body does not change, so we integrate the input rate over time and set it equal to the total amount in the body when the elimination rate equals the input rate. The steady state amount $A$ can be found by dividing the input rate by the elimination rate constant: $A = \frac{4 \text{ units/hour}}{1/12 \text{ hour}^{-1}} = 48 \text{ units}$.

A Hospital Patient Receives 4 Units of a Certain Drug f(t)=e−t/12f ( t ) = e ^ { - t / 12 }f(t)=e−t/12

A Hospital Patient Receives 4 Units of a Certain Drug $f ( t ) = e ^ { - t / 12 }$