A team of biologists is studying the evolution of the area covered by a marine algae called seagrass, on the bottom of Alycastre Bay, near the island of Porquerolles. The studied area has a total area of 20 hectares (ha), and on July 1, 2024, the seagrass covered 1 ha of this area.

\textbf{Part A: Study of a discrete model}

For any natural integer $n$, we denote by $u _ { n }$ the area of the zone, in hectares, covered by seagrass on July 1 of the year $2024 + n$. Thus, $u _ { 0 } = 1$. A study conducted on this area made it possible to establish that for any natural integer $n$:
$$u _ { n + 1 } = - 0{,}02 u _ { n } ^ { 2 } + 1{,}3 u _ { n }$$

\begin{enumerate}
  \item Calculate the area that seagrass should cover on July 1, 2025 according to this model.
  \item We denote by $h$ the function defined on $[0;20]$ by
  $$h ( x ) = - 0{,}02 x ^ { 2 } + 1{,}3 x$$
  We admit that $h$ is increasing on $[0;20]$.\\
  a. Prove that for any natural integer $n$, $1 \leqslant u _ { n } \leqslant u _ { n + 1 } \leqslant 20$.\\
  b. Deduce that the sequence $(u _ { n })$ converges. We denote by $L$ its limit.\\
  c. Justify that $L = 15$.
  \item The biologists wish to know after how long the area covered by seagrass will exceed 14 hectares.\\
  a. Without any calculation, justify that, according to this model, this will occur.\\
  b. Copy and complete the following algorithm so that at the end of execution, it displays the answer to the biologists' question.
\begin{verbatim}
def seuil():
    n=0
    u= 1
    while ...... :
        n=......
        u=......
    return n
\end{verbatim}
\end{enumerate}

\textbf{Part B: Study of a continuous model}

We wish to describe the area of the studied zone covered by seagrass over time with a continuous model. In this model, for a duration $t$, in years, elapsed from July 1, 2024, the area of the studied zone covered by seagrass is given by $f ( t )$, where $f$ is a function defined on $[ 0 ; + \infty [$ satisfying:
\begin{itemize}
  \item $f ( 0 ) = 1$;
  \item $f$ does not vanish on $[ 0 ; + \infty [$;
  \item $f$ is differentiable on $[ 0 ; + \infty [$;
  \item $f$ is a solution on $[ 0 ; + \infty [$ of the differential equation
  $$\left( E _ { 1 } \right) : \quad y ^ { \prime } = 0{,}02 y ( 15 - y ) .$$
\end{itemize}
We admit that such a function $f$ exists; the purpose of this part is to determine an expression for it. We denote by $f ^ { \prime }$ the derivative function of $f$.

\begin{enumerate}
  \item Let $g$ be the function defined on $\left[ 0 ; + \infty \left[ \text{ by } g ( t ) = \frac { 1 } { f ( t ) } \right. \right.$. Show that $g$ is a solution of the differential equation
  $$\left( E _ { 2 } \right) : \quad y ^ { \prime } = - 0{,}3 y + 0{,}02 .$$
  \item Give the solutions of the differential equation $( E _ { 2 } )$.
  \item Deduce that for all $t \in [ 0 ; + \infty [$:
  $$f ( t ) = \frac { 15 } { 14 \mathrm { e } ^ { - 0{,}3 t } + 1 }$$
  \item Determine the limit of $f$ as $+ \infty$.
  \item Solve in the interval $[ 0 ; + \infty [$ the inequality $f ( t ) > 14$. Interpret the result in the context of the exercise.
\end{enumerate}