Science:Math Exam Resources/Courses/MATH110/December 2015/Question 07 (c)

MATH110 December 2015

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[hide] Question 07 (c)

An exponential decay function $P(t)=Ce^{-kt}$, where $C$ and $k$ are positive constants, is used to model the amount of drug in the blood t hours after an initial dose is administered. Suppose an initial dose of 100 mg is administered to a patient. Assume it takes 36 hr for the body to eliminate half of the initial dose from the blood stream. Answer the questions below. You may leave your answers unsimplified, i.e. in ”calculator-ready” form. (c) If a second 100-mg dose is given 36 hr after the first dose, how much time (since the first dose) is required for the drug-level in the blood stream to reach 1 mg?

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[show] Hint
What is the remaining amount of drug from the first dose after 36 hours? Adding a second dose of another 100mg, what is the amount of drug at $t=36$ hrs?

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[show] Solution

By the given information in the question, if the initial drug is 100 mg after 36 hours there will be 50 mg, now adding another 100 mg the new initial amount is 150 mg, so the first step is to find how long it takes to start with 150 mg and end up with 1 mg. By the model, this time we have $C=150$ (initial dose), and hence $P(t)=150e^{-{\frac {\ln 2}{36}}t}$. Here, $t$ represents the time from the second dose. To get the time at which the amount of drug becomes 1 mg, we solve $P(t)=1\iff {\frac {1}{150}}=e^{-{\frac {\ln 2}{36}}t}\iff \ln(150)={\frac {\ln 2}{36}}t\iff t={\frac {36\ln 150}{\ln 2}}$ , Therefore, it takes ${\frac {36\ln 150}{\ln 2}}$ hrs from the second dose, therefore, the total time from the first dose is $\color {blue}t=36+{\frac {36\ln 150}{\ln 2}}\ hrs$