Answer
See the full explanation below.
Work Step by Step
(a)
The value of the number of discharges for ${{x}_{1}}=7$ is:
$\begin{align}
& f\left( x \right)=1.1{{x}^{3}}-35{{x}^{2}}+264x+557 \\
& f\left( 7 \right)=1.1{{\left( 7 \right)}^{3}}-35{{\left( 7 \right)}^{2}}+264\left( 7 \right)+557 \\
& f\left( 7 \right)=377.3-1715+1848+557 \\
& f\left( 7 \right)=1067.3 \\
\end{align}$
The value of the number of discharges for ${{x}_{2}}=12$ is
$\begin{align}
& f\left( x \right)=1.1{{x}^{3}}-35{{x}^{2}}+264x+557 \\
& f\left( 12 \right)=1.1{{\left( 12 \right)}^{3}}-35{{\left( 12 \right)}^{2}}+264\left( 12 \right)+557 \\
& f\left( 12 \right)=1900.8-5040+3168+557 \\
& f\left( 12 \right)=585.8
\end{align}$
Substitute the values $\left( {{x}_{2}},{{x}_{1}} \right)=\left( 12,7 \right)$and $\left( f\left( {{x}_{2}} \right),f\left( {{x}_{1}} \right) \right)=\left( 585.8,1067.3 \right)$ to get the rate of change:
$\begin{align}
& m=\frac{f\left( {{x}_{2}} \right)-f\left( {{x}_{1}} \right)}{{{x}_{2}}-{{x}_{1}}} \\
& =\frac{585.5-1067.3}{12-7} \\
& =\frac{-481.8}{5} \\
& \approx -96
\end{align}$
Therefore, the slope of the secant line is $-96$ from ${{x}_{1}}=7\text{ to }{{x}_{2}}=12$.
(b)
The difference between the slope determines whether it underestimates or overestimates.
Difference in slope is
$\begin{align}
& {{m}_{2}}-{{m}_{1}}=-130-\left( -96 \right) \\
& \Delta m=-34
\end{align}$
Hence, the slope underestimates the decrease by $34$ discharges per year.
