\textbf{Exercise 4 (Candidates who have not followed the specialization course)}

The common spruce is a species of coniferous tree that can measure up to 40 meters in height and live more than 150 years. The objective of this exercise is to estimate the age and height of a spruce based on the diameter of its trunk measured at $1.30 \mathrm {~m}$ from the ground.

\section*{Part A - Modeling the age of a spruce}
For a spruce whose age is between 20 and 120 years, the relationship between its age (in years) and the diameter of its trunk (in meters) measured at $1.30 \mathrm {~m}$ from the ground is modeled by the function $f$ defined on the interval $] 0 ; 1 [$ by:
$$f ( x ) = 30 \ln \left( \frac { 20 x } { 1 - x } \right)$$
where $x$ denotes the diameter expressed in meters and $f ( x )$ the age in years.

\begin{enumerate}
  \item Prove that the function $f$ is strictly increasing on the interval $] 0 ; 1 [$.
  \item Determine the values of the trunk diameter $x$ such that the age calculated in this model remains consistent with its validity conditions, that is, between 20 and 120 years.
\end{enumerate}

\section*{Part B}
The average height of spruces in representative samples of trees aged 50 to 150 years was measured. The following table, created using a spreadsheet, groups these results and allows calculation of the average growth rate of a spruce.

\begin{center}
\begin{tabular}{|l|l|l|l|l|l|l|l|l|l|l|l|l|l|}
\hline
 & A & B & C & D & E & F & G & H & I & J & K & L & M \\
\hline
1 & Ages (in years) & 50 & 70 & 80 & 85 & 90 & 95 & 100 & 105 & 110 & 120 & 130 & 150 \\
\hline
2 & Heights (in meters) & 11.2 & 15.6 & 18.05 & 19.3 & 20.55 & 21.8 & 23 & 24.2 & 25.4 & 27.6 & 29.65 & 33 \\
\hline
3 & Growth rate (in meters per year) &  & 0.22 & 0.245 & 0.25 &  &  &  &  &  &  &  &  \\
\hline
\end{tabular}
\end{center}

\begin{enumerate}
  \item a. Interpret the number 0.245 in cell D3.\\
b. What formula should be entered in cell C3 to complete line 3 by copying cell C3 to the right?
  \item Determine the expected height of a spruce whose trunk diameter measured at $1.30 \mathrm {~m}$ from the ground is 27 cm.
  \item The quality of the wood is better when the growth rate is maximal.\\
a. Determine an age interval during which the wood quality is best by explaining the approach.\\
b. Is it consistent to ask loggers to cut trees when their diameter measures approximately 70 cm?
\end{enumerate}