A hospital patient receives intravenously 15 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 $f ( t ) = e ^ { - t / 6 }$ .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)90 units
B)2.50 units
C)75 units
D)21 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 of the drug infusion and elimination. The rate of drug infusion is 15 units per hour. The elimination rate is given by the function $f(t) = e^{-t/6}$, which means that the fraction of the drug eliminated increases exponentially with time. To find the steady-state amount, we set the infusion rate equal to the elimination rate. However, since the drug is eliminated exponentially, we need to find the amount that accumulates over time, not just the balance of infusion and elimination at a specific moment. The steady-state occurs when the amount of drug entering the body equals the amount being eliminated over the same period. The formula for the steady-state amount (S) is given by the infusion rate (I) divided by the elimination rate constant (k), where k is derived from the half-life or the elimination function. From the given function, $k = \frac{1}{6}$ per hour because the drug's half-life is related to the exponent in the elimination function. Thus, $S = \frac{I}{k} = \frac{15}{1/6} = 15 \times 6 = 90$ units. This calculation assumes a continuous process where the rate of drug input equals the rate of drug elimination at steady state.

A Hospital Patient Receives Intravenously 15 Units of a Certain f(t)=e−t/6f ( t ) = e ^ { - t / 6 }f(t)=e−t/6

A Hospital Patient Receives Intravenously 15 Units of a Certain $f ( t ) = e ^ { - t / 6 }$