We denote $\mathscr { A } ( \theta )$ the region bounded by the lines with equations $t = 10 , t = \theta$, $y = 85$ and the curve $\mathscr { C } _ { f }$ representing $f$. We consider that sterilization is complete after a time $\theta$, if the area, expressed in square units of the region $\mathscr { A } ( \theta )$ is greater than 80.
[a.] Justify, using the graph given in the appendix, that we have $\mathscr { A } ( 25 ) > 80$.
[b.] Justify that, for $\theta \geqslant 10$, we have $\mathscr { A } ( \theta ) = 15 ( \theta - 10 ) - 75 \int _ { 10 } ^ { \theta } \mathrm { e } ^ { - \frac { \ln 5 } { 10 } t } \mathrm {~d} t$.
[c.] Is sterilization complete after 20 minutes?
We denote $\mathscr { A } ( \theta )$ the region bounded by the lines with equations $t = 10 , t = \theta$, $y = 85$ and the curve $\mathscr { C } _ { f }$ representing $f$.\\
We consider that sterilization is complete after a time $\theta$, if the area, expressed in square units of the region $\mathscr { A } ( \theta )$ is greater than 80.
\begin{enumerate}
\item[a.] Justify, using the graph given in the appendix, that we have $\mathscr { A } ( 25 ) > 80$.
\item[b.] Justify that, for $\theta \geqslant 10$, we have $\mathscr { A } ( \theta ) = 15 ( \theta - 10 ) - 75 \int _ { 10 } ^ { \theta } \mathrm { e } ^ { - \frac { \ln 5 } { 10 } t } \mathrm {~d} t$.
\item[c.] Is sterilization complete after 20 minutes?
\end{enumerate}