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.
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