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

MATH110 December 2015

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Question 07 (a)

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. (a) Find the exponential function that predicts the amount of drug $P(t)$ in the blood at $t$ hr. Make sure your answer has no unknown constants.

Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you?

If you are stuck, check the hint below. Consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it!

Hint
You need to fill the given model with the constant $C$ which is the initial quantity and the constant $k$ which can be calculated with the given info in the question.

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Solution

Found a typo? Is this solution unclear? Let us know here. Please rate my easiness! It's quick and helps everyone guide their studies. The initial amount of drug is given by 100 mg, thus $C=100$ mg. We also know that it takes 36 hr to have half of the initial dose remaining in the blood, i.e. $T_{1/2}=36$ hr, on the other hand, we have the formula that $k={\frac {\ln 2}{T_{1/2}}}={\frac {\ln 2}{36}}$; Indeed, it follows from ${\frac {1}{2}}\cdot (100)=P(36)=100e^{-36k}\implies {\frac {1}{2}}=e^{-36k}\implies -\ln 2=-36k\implies k={\frac {\ln 2}{36}}.$ Therefore, the model will be $\color {blue}P(t)=100e^{-{\frac {\ln 2}{36}}t}$