To determine when the patient will first need to take additional medicine, we can use the given information about the medication level decreasing by 27.5% every hour.
Let the initial medicine level be represented by M, and the level greater than 100 milligrams be 100. Thus, the initial medicine level is greater than 100 milligrams.
If the medicine level decreases by 27.5% every hour, this can be represented by the equation:
M = M - 0.275M
Simplifying this equation, we get:
M = 0.725M
Since we want to find the time when the medicine level will drop below 100 milligrams, we can set up the following inequality:
0.725^n * M > 100
0.725^n * M > 100
0.725^n * M > 100
0.725^n * M > 100
To solve for n (hours), we can divide both sides by M:
0.725^n > 100 / M
n > log(100 / M) / log(0.725)
Since M > 100, we can substitute M = 100 into the equation:
n > log(100 / 100) / log(0.725)
n > log(1) / log(0.725)
n > 0 / -0.1392
n < 0
Therefore, the patient will need to take additional medicine after 0 hours, meaning the patient would need to take additional medicine immediately.
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.
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