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After a certain medicine is injected, its concentration in the bloodstream changes exponentially over time, The graph describes the medicine's concentration (in milligrams per liter) over time (in hours).

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Betti · Answer 1 · 2023-07-25T01:49:46+0000

Answer:

13.79 hours

Explanation:

The decay model for the medication is:

$M=400\left(0.86\right)^t$

When the initial amount decreases to 50 mg:

$50=400\left(0.86\right)^t$

We then solve the equation for t:

$\begin{gathered} \text{ Divide both sides by 400} \\ (50)/(400)=(400)/(400)\left(0.86\right)^t \\ (1)/(8)=0.86^t \\ 0.125=0.86^t \\ \text{ Take the log of both sides} \\ \log(0.125)=\log(0.86^t) \\ \operatorname{\log}(0.125)=t\operatorname{\log}(0.86) \\ \text{ Divide both sides by }\log0.86 \\ \frac{\begin{equation*}t\operatorname{\log}(0.86)\end{equation*}}{\operatorname{\log}(0.86)}=\frac{\operatorname{\log}(0.125)}{\operatorname{\log}(0.86)} \\ t\approx13.79\text{ hours} \end{gathered}$

It will take 13.79 hours for the initial 400 mg of medicine to decrease to 50 mg.

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