A company manufactures plastic objects by injecting molten material at $210 ^ { \circ } \mathrm { C }$ into a mold. We seek to model the cooling of the material using a function $f$ giving the temperature of the injected material as a function of time $t$.\\
Time is expressed in seconds and temperature is expressed in degrees Celsius.\\
We assume that the function $f$ sought is a solution to a differential equation of the following form, where $m$ is a real constant that we seek to determine:

$$( E ) : \quad y ^ { \prime } + 0,02 y = m$$

\textbf{Part A}
\begin{enumerate}
  \item Justify the following output from a computer algebra system:
\end{enumerate}

\begin{center}
\begin{tabular}{ | l | l | }
\hline
Input: & SolveDifferentialEquation $\left( y ^ { \prime } + 0,02 y = m \right)$ \\
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Output: & $\rightarrow y = k * \exp ( - 0.02 * t ) + 50 * m$ \\
\hline
\end{tabular}
\end{center}

\begin{enumerate}
  \setcounter{enumi}{1}
  \item The workshop temperature is $30 ^ { \circ } \mathrm { C }$. We assume that the temperature $f ( t )$ tends toward $30 ^ { \circ } \mathrm { C }$ as $t$ tends toward infinity.\\
Prove that $m = 0,6$.
  \item Determine the expression of the function $f$ sought, taking into account the initial condition $f ( 0 ) = 210$.
\end{enumerate}

\textbf{Part B}\\
We assume here that the temperature (expressed in degrees Celsius) of the injected material as a function of time (expressed in seconds) is given by the function whose expression and graphical representation are given below:

$$f ( t ) = 180 \mathrm { e } ^ { - 0,02 t } + 30$$

\begin{enumerate}
  \item The object can be removed from the mold when its temperature becomes less than $50 ^ { \circ } \mathrm { C }$.\\
a. By graphical reading, give an approximate value of the number $T$ of seconds to wait before removing the object from the mold.\\
b. Determine by calculation the exact value of this time $T$.
  \item Using an integral, calculate the average value of the temperature over the first 100 seconds.
\end{enumerate}