Question

At the moment a certain medicine is injected, its concentration in the bloodstream is
\[120\] milligrams per liter. From that moment forward, the medicine's concentration drops by
\[30\%\] each hour.
Write a function that gives the medicine's concentration in milligrams per liter,
\[C(t)\],
\[t\] hours after the medicine was injected.
\[C(t)=\]

Answer 1

To model the concentration of the medicine in the bloodstream over time, we can use the formula for exponential decay. The concentration drops by \(30\%\) each hour, which means that \(70\%\) of the concentration remains after each hour.

Initially, the concentration \(C(0)\) is \(120\) milligrams per liter. After \(t\) hours, the concentration can be calculated as follows:

\[
C(t) = C(0) \times (1 - r)^t
\]

where \(r\) is the decay rate. Given that the concentration drops by \(30\%\), we have:

\[
r = 0.30
\]

Thus, \(1 - r = 0.70\). Plugging in the initial concentration:

\[
C(t) = 120 \times (0.70)^t
\]

This gives us the function for the concentration of the medicine in the bloodstream \(t\) hours after the injection:

\[
C(t) = 120 \times (0.70)^t
\]

This equation will allow you to find the concentration of the medicine at any given hour \(t\) after injection.

Answer 2

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