A treatment protocol for a disease in children involves long-term infusion of an appropriate medication. The concentration of the medication in the blood over time is modeled by the function $C$ defined on the interval $[0; +\infty[$ by:

$$C ( t ) = \frac { d } { a } \left( 1 - \mathrm { e } ^ { - \frac { a } { 80 } t } \right)$$

where $C$ denotes the concentration of the medication in the blood (in micromoles per liter), $t$ the time elapsed since the start of the infusion (in hours), $d$ the infusion rate (in micromoles per hour), $a$ a strictly positive real parameter called clearance (in liters per hour).

\textbf{Part C: determination of appropriate treatment}

The purpose of this part is to determine, for a given patient, the value of the infusion rate that allows the treatment to be effective, that is, the plateau to equal 15. The infusion rate $d$ is provisionally set to 105.

\begin{enumerate}
  \item We seek to determine the clearance $a$ of a patient. The infusion rate is provisionally set to 105.\\
a. Express as a function of $a$ the concentration of the medication 6 hours after the start of the infusion.\\
b. After 6 hours, analyses allow us to know the concentration of the medication in the blood; it is equal to 5.9 micromoles per liter.\\
Determine an approximate value, to the nearest tenth of a liter per hour, of the clearance of this patient.
  \item Determine the value of the infusion rate $d$ guaranteeing the effectiveness of the treatment.
\end{enumerate}