Writing functions with exponential decay
Google Classroom
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
Therefore, the function that gives the medicine's concentration in milligrams per liter, \(C(t)\), \(t\) hours after the medicine was injected is \(\boxed{C(t) = 120 \cdot e^{-0.3t}}\).
Step 1 ：The problem describes an exponential decay situation. The general form of an exponential decay function is \(C(t) = C_0 \cdot e^{-kt}\), where \(C_0\) is the initial concentration, \(k\) is the decay constant, and \(t\) is the time.
Step 2 ：In this case, the initial concentration \(C_0\) is 120 milligrams per liter, and the decay rate is 30%, or 0.3. So, the decay constant \(k\) is 0.3.
Step 3 ：Substitute \(C_0 = 120\) and \(k = 0.3\) into the general form of the exponential decay function, we get \(C(t) = 120 \cdot e^{-0.3t}\).
Step 4 ：Therefore, the function that gives the medicine's concentration in milligrams per liter, \(C(t)\), \(t\) hours after the medicine was injected is \(\boxed{C(t) = 120 \cdot e^{-0.3t}}\).
