Answer
a) $768.79$ gallons
b) $933.10$ gallons
c) $681.51$ gallons
Work Step by Step
Given \begin{equation}
F(t)=0.425 t^2-16.431 t+840.321.
\end{equation} a) Set $t= 5$ to estimate the average fuel consumption in $1975$.
\begin{equation}
\begin{aligned}
F(5)=0.425 (5)^2-16.431 (5)+840.321 =768.79.
\end{aligned}
\end{equation} The average fuel consumption in $1975$ was about $768.79$ gallons per vehicle.
b)Set $t= -5$ to estimate the average fuel consumption in $1985$.
\begin{equation}
\begin{aligned}
F(-5)=0.425 (-5)^2-16.431 (-5)+840.321 = 933.10.
\end{aligned}
\end{equation} The average fuel consumption in $1985$ was about $933.10$ gallons per vehicle.
c) The vertex of the function will give us the minimum fuel consumption per vehicle per year .Use $a= 0.425$ and $b= -16.431$ into the following formula.
$$
\begin{aligned}
t & =\frac{-b}{2 a} \\
& =\frac{-(-16.43)}{2(0.425)} \\
& =19.33.
\end{aligned}
$$ $$
\begin{aligned}
F_{min} & =0.425(19.33)^2-16.431(19.33)+840.321 \\
& =681.51.
\end{aligned}
$$ The vertex is $(19.33,681.51)$. This means that the lowest fuel consumption per vehicle occurred in $1994$ and was about $681.51$ gallons per vehicle.
