Answer
a)
The whale population will reach 15,000 in about 22 years after 1982, in 2004.
b)
The whale population will reach 25,000 in about 33 years after 1982, in 2014.
c)
The graph is shown below.
Work Step by Step
a)
To calculate the year when the whale population will reach 15,000, substitute $x=15$ in $Y\left( x \right)=21.77\ln \frac{x}{5.5}$:
$\begin{align}
& Y\left( x \right)=21.77\ln \frac{x}{5.5} \\
& Y\left( x \right)=21.77\ln \frac{15}{5.5}
\end{align}$
Using a calculator, $\ln \frac{15}{5.5}\approx 1.00330$. Therefore,
$\begin{align}
& Y\left( x \right)=21.77\times 1.00330 \\
& \approx 21.84 \\
& \approx 22
\end{align}$
b)
To calculate the year when the whale population will reach 25,000, substitute $x=25$ in $Y\left( x \right)=21.77\ln \frac{x}{5.5}$:
$\begin{align}
& Y\left( x \right)=21.77\ln \frac{x}{5.5} \\
& Y\left( x \right)=21.77\ln \frac{25}{5.5}
\end{align}$
$\ln \frac{25}{5.5}\approx 1.51412$. Therefore,
$\begin{align}
& Y\left( x \right)=21.77\times 1.51412 \\
& \approx 32.96 \\
& \approx 33
\end{align}$
c)
From the above parts,
$Y\left( x \right)=22$, when $x=15$ (from part a)
$Y\left( x \right)=33$, when $x=25$(from part b)
Also, when $x=5.5$,
$\begin{align}
& Y\left( x \right)=21.77\ln \frac{x}{5.5} \\
& Y\left( x \right)=21.77\ln \frac{5.5}{5.5} \\
& =21.77\ln 1
\end{align}$
$\ln 1=0$. Therefore,
$\begin{align}
& Y\left( x \right)=21.77\times 0 \\
& =0
\end{align}$
When $x=10$,
$\begin{align}
& Y\left( x \right)=21.77\ln \frac{x}{5.5} \\
& Y\left( x \right)=21.77\ln \frac{10}{5.5}
\end{align}$
$\ln \frac{10}{5.5}\approx 0.597$. Therefore,
$\begin{align}
& Y\left( x \right)\approx 21.77\times 0.597 \\
& \approx 12.99 \\
& \approx 13
\end{align}$
When $x=12$,
$\begin{align}
& Y\left( x \right)=21.77\ln \frac{x}{5.5} \\
& Y\left( x \right)=21.77\ln \frac{12}{5.5}
\end{align}$
$\ln \frac{12}{5.5}\approx 0.78$. Therefore,
$\begin{align}
& Y\left( x \right)\approx 21.77\times 0.78 \\
& \approx 16.9
\end{align}$
Plot the following points on the graph:
$\begin{matrix}
x & y \\
15 & 22 \\
25 & 33 \\
5.5 & 0 \\
10 & 13 \\
12 & 16.9 \\
\end{matrix}$
