The rate (as a percentage) of $\mathrm{CO}_2$ contained in a room after $t$ minutes of hood operation is modelled by the function $f$ defined for all real $t$ in the interval $[0;20]$ by: $$f(t) = (0{,}8t + 0{,}2)\mathrm{e}^{-0{,}5t} + 0{,}03.$$ It is desired that the rate of $\mathrm{CO}_2$ in the room returns to a value $V$ less than or equal to $3.5\%$. a. Justify that there exists a unique instant $T$ satisfying this condition. b. Consider the following algorithm: \begin{verbatim} $t \leftarrow 1,75$ $p \leftarrow 0,1$ $V \leftarrow 0,7$ While $V > 0,035$ $t \leftarrow t + p$ $V \leftarrow ( 0,8 t + 0,2 ) \mathrm { e } ^ { - 0,5 t } + 0,03$ End While \end{verbatim} What is the value of the variable $t$ at the end of the algorithm? What does this value represent in the context of the exercise?
The rate (as a percentage) of $\mathrm{CO}_2$ contained in a room after $t$ minutes of hood operation is modelled by the function $f$ defined for all real $t$ in the interval $[0;20]$ by:
$$f(t) = (0{,}8t + 0{,}2)\mathrm{e}^{-0{,}5t} + 0{,}03.$$
It is desired that the rate of $\mathrm{CO}_2$ in the room returns to a value $V$ less than or equal to $3.5\%$.\\
a. Justify that there exists a unique instant $T$ satisfying this condition.\\
b. Consider the following algorithm:
\begin{verbatim}
$t \leftarrow 1,75$
$p \leftarrow 0,1$
$V \leftarrow 0,7$
While $V > 0,035$
$t \leftarrow t + p$
$V \leftarrow ( 0,8 t + 0,2 ) \mathrm { e } ^ { - 0,5 t } + 0,03$
End While
\end{verbatim}
What is the value of the variable $t$ at the end of the algorithm? What does this value represent in the context of the exercise?