Total personal income of the country $($in billions of dollars$)$ for selected years from $1955$ to $2000$ is given in the table.

$($a$)$ These data can be modeled by an exponential function. Write the equation of this function, with $x$ as the number of years after $1955.$

$($b$)$ If this model is accurate, what will be the country's total personal income in $2005?$

$($c$)$ In what year does the model predict the total personal income will reach $$23$ trillion?

$\backslash$table$[[$Year$,$Personal Income$],[1955,411\mathrm{.}9],[1965,833\mathrm{.}8],[1975,2312\mathrm{.}2],[1985,4881\mathrm{.}6],[1995,8429\mathrm{.}1],[2000,10,231\mathrm{.}6$