Suppose that the level of medicine in the patient's body must maintain a level greater than 100 milligrams. After how many hours will this patient first need to take the additional medicine? Show your work or explain your answer to receive credit for this question.

Answer 1

To determine after how many hours the patient will first need to take additional medicine, we can use the formula for exponential decay:

\[P(t) = P_0 \cdot e^{-kt}\]

where:
- \(P(t)\) = the amount of medicine in the patient's body after \(t\) hours
- \(P_0\) = the initial medicine level
- \(k\) = decay constant
- \(t\) = time in hours

Given that the initial medicine level is 200 milligrams and the medicine must maintain a level greater than 100 milligrams, we can set up the following inequality:

\[P(t) > 100\]

Substitute the formula for exponential decay into the inequality:

\[200 \cdot e^{-kt} > 100\]

Solve for \(t\):

\[e^{-kt} > \frac{100}{200}\]

\[e^{-kt} > \frac{1}{2}\]

\[{-kt} > \ln{\frac{1}{2}}\]

\[t < -\frac{\ln{(\frac{1}{2})}}{k}\]

Since we do not have the specific decay constant \(k\), we cannot determine the exact amount of time in hours. However, after the patient takes the initial medicine, they will need to take additional medicine before the medicine level drops below 100 milligrams.