Answer
The system income will be $\$1177$ billion in 14 years after 2004 or 2018.
Work Step by Step
Considered the formula,
$B=0.07{{x}^{2}}+47.4x+500$,
Now, the amount paid in benefits will be $\$1177$ billion.
So,
$1177=0.07{{x}^{2}}+47.4x+500$
Simplified,
$\begin{align}
& 0.07{{x}^{2}}+47.4x+500-1177=0 \\
& 0.07{{x}^{2}}+47.4x-677=0
\end{align}$
So, this is a quadratic equation
Here, $a=0.07,\text{ }b=47.4\text{ and }c=-677$
Now,
$\begin{align}
& x=\frac{-\left( 47.4 \right)\pm \sqrt{{{\left( 47.4 \right)}^{2}}-4\left( 0.07 \right)\left( -677 \right)}}{2\left( 0.07 \right)} \\
& =\frac{-47.4\pm \sqrt{2246.76+189.56}}{0.14} \\
& =\frac{-47.4\pm \sqrt{2436.32}}{0.14} \\
& =\frac{-47.4\pm 49.36}{0.14}
\end{align}$
Further simplified,
$\begin{align}
& x=\frac{-47.4+49.36}{0.14} \\
& =\frac{1.96}{0.14} \\
& =14
\end{align}$
Or,
$\begin{align}
& x=\frac{-47.4-49.36}{0.14} \\
& =\frac{-96.76}{0.14} \\
& =-691.143
\end{align}$
Take only the positive value, so, $x=14$.
Hence, the system income will be $\$1177$ billion in 14 years after 2004 or 2018.