Alain owns a swimming pool that contains $50\mathrm{~m}^3$ of water. Recall that $1\mathrm{~m}^3 = 1000\mathrm{~L}$. To disinfect the water, he must add chlorine. The chlorine level in the water, expressed in $\mathrm{mg}\cdot\mathrm{L}^{-1}$, is defined as the mass of chlorine per unit volume of water. Pool specialists recommend a chlorine level between 1 and $3\mathrm{~mg}\cdot\mathrm{L}^{-1}$. Under the action of the ambient environment, particularly ultraviolet rays, chlorine decomposes and gradually disappears. On Wednesday, June 19, he measures a chlorine level of $0.70\mathrm{~mg}\cdot\mathrm{L}^{-1}$.

\textbf{Part A: study of a discrete model.}

To maintain the chlorine level in his pool, Alain decides, starting from Thursday, June 20, to add 15 g of chlorine each day. It is admitted that this chlorine mixes uniformly in the pool water.

\begin{enumerate}
  \item Justify that this addition of chlorine increases the level by $0.3\mathrm{~mg}\cdot\mathrm{L}^{-1}$.
  \item For any natural number $n$, we denote by $v_n$ the chlorine level, in $\mathrm{mg}\cdot\mathrm{L}^{-1}$, obtained with this new protocol $n$ days after Wednesday, June 19. Thus $v_0 = 0.7$. It is admitted that for any natural number $n$,
$$v_{n+1} = 0.92 v_n + 0.3.$$
a. Show by induction that for any natural number $n$, $\quad v_n \leqslant v_{n+1} \leqslant 4$.\\
b. Show that the sequence $(v_n)$ is convergent and calculate its limit.
  \item In the long term, will the chlorine level comply with the pool specialists' recommendation? Justify your answer.
  \item Reproduce and complete the algorithm below written in Python language so that the function \texttt{alerte\_chlore} returns, when it exists, the smallest integer $n$ such that $v_n > s$.
\begin{verbatim}
def alerte_chlore(s) :
    n = 0
    u=0.7
    while...:
        n = ...
        u=...
    return n
\end{verbatim}
  \item What value is obtained by entering the instruction \texttt{alerte\_chlore(3)}? Interpret this result in the context of the exercise.
\end{enumerate}

\textbf{Part B: study of a continuous model.}

Alain decides to call on a specialized engineering firm. This firm uses a continuous model to describe the chlorine level in the pool. In this model, for a duration $x$ (in days elapsed from Wednesday, June 19), $f(x)$ represents the chlorine level, in $\mathrm{mg}\cdot\mathrm{L}^{-1}$, in the pool. It is admitted that the function $f$ is a solution of the differential equation $(E): y' = -0.08y + \frac{q}{50}$, where $q$ is the quantity of chlorine, in grams, added to the pool each day.

\begin{enumerate}
  \item Justify that the function $f$ is of the form $f(x) = C\mathrm{e}^{-0.08x} + \frac{q}{4}$ where $C$ is a real constant.
  \item a. Express as a function of $q$ the limit of $f$ at $+\infty$.\\
b. Recall that the chlorine level observed on Wednesday, June 19 is equal to $0.7\mathrm{~mg}\cdot\mathrm{L}^{-1}$. It is desired that the chlorine level stabilizes in the long term around $2\mathrm{~mg}\cdot\mathrm{L}^{-1}$. Determine the values of $C$ and $q$ so that these two conditions are satisfied.
\end{enumerate}