Answer
a) $\$791.76$
b) $1990$ and $2001$
Work Step by Step
Given \begin{equation}
B(t)=\frac{-470001 t^2+4110992 t+14032612}{-469.4 t^2+3745 t+19774}.
\end{equation} The year $2000$ corresponds to $t= 10$ since $1990$.
a) Find $B(10)$: \begin{equation}
\begin{aligned}
B(10)&= \frac{-470001\cdot 10^2+4110992 \cdot 10+14032612}{-469.4 \cdot 10^2+3745\cdot 10+19774}\\
&=\frac{8142432}{10284}\\
&\approx 791.76.
\end{aligned}
\end{equation} The average benefit for a person participating in the U.S. food stamp program in $2000$ was about $\$791.76$.
b) Set $B(t) = 700$ and graph the right and left hand side functions in the same window. \begin{equation}
\begin{aligned}
B(t) & = 700\\
\frac{-470001 t^2+4110992 t+14032612}{-469.4 t^2+3745 t+19774}&= 700.
\end{aligned}
\end{equation} Let \begin{equation}
\begin{aligned}
f(t)& =B(t)\\
g(t)&= 700.
\end{aligned}
\end{equation} The points of intersection between the two functions occur at about $t= 0$ and about $t= 10.65$. Hence, the average benefit for a person participating in the U.S. food stamp program was approximately $\$700$ in $1990$ and again in early $2001$.
