In the Pyrenees National Park, a researcher is working on the decline of a protected species in high-mountain lakes: the ``midwife toad''.\\
Parts I and II can be approached independently.

\section*{Part I: Effect of the introduction of a new species}
In certain lakes in the Pyrenees, trout have been introduced by humans to enable fishing activities in the mountains. The researcher studied the impact of this introduction on the midwife toad population in a lake.\\
His previous studies lead him to model the evolution of this population as a function of time by the following function $f$:
$$f ( t ) = \left( 0.04 t ^ { 2 } - 8 t + 400 \right) \mathrm { e } ^ { \frac { t } { 50 } } + 40 \text { for } t \in [ 0 ; 120 ]$$
The variable $t$ represents the elapsed time, in days, from the introduction at time $t = 0$ of trout into the lake, and $f ( t )$ models the number of toads at time $t$.

\begin{enumerate}
  \item Determine the number of toads present in the lake when the trout are introduced.
  \item We admit that the function $f$ is differentiable on the interval $[0 ; 120]$ and we denote $f ^ { \prime }$ its derivative function.\\
Show, by displaying the calculation steps, that for every real number $t$ belonging to the interval $[0 ; 120]$ we have:
$$f ^ { \prime } ( t ) = t ( t - 100 ) \mathrm { e } ^ { \frac { t } { 50 } } \times 8 \times 10 ^ { - 4 }$$
  \item Study the variations of the function $f$ on the interval $[0 ; 120]$, then draw up the variation table of $f$ on this interval (approximate values to the nearest hundredth will be given).
  \item According to this model:\\
a. Determine the number of days $J$ necessary for the number of toads to reach its minimum. What is this minimum number?\\
b. Justify that, after reaching its minimum, the number of toads will one day exceed 140 individuals.\\
c. Using a calculator, determine the duration in days from which the number of toads will exceed 140 individuals.
\end{enumerate}

\section*{Part II: Effect of Chytridiomycosis on a tadpole population}
One of the main causes of the decline of this toad species in high mountains is a disease, ``Chytridiomycosis'', caused by a fungus.\\
The researcher considers that:
\begin{itemize}
  \item Three quarters of the mountain lakes in the Pyrenees are not infected by the fungus, that is, they contain no contaminated tadpoles (toad larvae).
  \item In the remaining lakes, the probability that a tadpole is contaminated is 0.74.
\end{itemize}
The researcher randomly chooses a lake in the Pyrenees and takes samples from it.\\
For the rest of the exercise, results will be rounded to the nearest thousandth when necessary. The researcher randomly takes a tadpole from the chosen lake to perform a test before releasing it.\\
We denote $T$ the event ``The tadpole is contaminated by the disease'' and $L$ the event ``The lake is infected by the fungus''.\\
We denote $\bar { L }$ the opposite event of $L$ and $\bar { T }$ the opposite event of $T$.

\begin{enumerate}
  \item Copy and complete the following probability tree using the data from the problem statement.
  \item Show that the probability $P ( T )$ that the sampled tadpole is contaminated is 0.185.
  \item The tadpole is not contaminated. What is the probability that the lake is infected?
\end{enumerate}