Dosing interval for a decaying drug concentration

After one dose of a medicine, the concentration in a patient’s blood is modelled by \(C=12e^{-0.2t}\) mg/L, where \(t\) is the time in hours after the dose. The medicine is effective only while \(C\ge 3\). Assume that when a second dose is taken, its concentration adds to whatever remains.

1. Find, in the form \(k\ln m\), the longest time after one dose for which the medicine remains effective.[3]
2. A second dose is taken \(6\) hours after the first. Show that the concentration immediately afterwards is \(12\left(1+e^{-1.2}\right)\) mg/L, and find how long after the second dose the medicine remains effective.[4]
3. Explain why the answer to (b) is longer than the answer to (a), and state one way in which the model may be unrealistic.[2]