Total personal income of the country (in billions of dollars) for selected years from 1957 to 2002 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 1957 .
(b) If this model is accurate, what will be the country's total personal income in 2007?
(c) In what year does the model predict the total personal income will reach $\$ 22$ trillion?
\begin{tabular}{|c|c|}
\hline Year & \begin{tabular}{c}
Personal \\
Income
\end{tabular} \\
\hline 1957 & 410.5 \\
\hline 1967 & 838.3 \\
\hline 1977 & 2309.6 \\
\hline 1987 & 4884.2 \\
\hline 1997 & 8429.5 \\
\hline 2002 & $10,237.6$ \\
\hline
\end{tabular}
(a) The equation of an exponential function that models the data is $y=\square$. (Use integers or decimals for any numbers in the expression. Round to three decimal places as needed.)

Step 1 ：The problem is asking to find an exponential function that models the given data. An exponential function has the form \(y = ab^x\), where \(a\) is the initial value (the value of \(y\) when \(x=0\)), \(b\) is the growth factor, and \(x\) is the exponent. In this case, \(x\) represents the number of years after 1957 and \(y\) represents the total personal income of the country in billions of dollars.

Step 2 ：To find the values of \(a\) and \(b\), we can use two data points from the table. Let's use the data for the years 1957 and 1967. In 1957, \(x=0\) and \(y=410.5\). In 1967, \(x=10\) and \(y=838.3\).

Step 3 ：We can set up two equations using these data points and solve for \(a\) and \(b\). The first equation is \(410.5 = ab^0\), which simplifies to \(410.5 = a\).

Step 4 ：The second equation is \(838.3 = a*b^{10}\). Substituting \(a=410.5\) into this equation, we get \(838.3 = 410.5*b^{10}\). Solving this equation for \(b\) will give us the growth factor.

Step 5 ：By solving the equation, we find that the growth factor \(b\) is approximately 1.074. Therefore, the equation of the exponential function that models the data is \(y = 410.5*(1.074)^x\).

Step 6 ：\(\boxed{y = 410.5*(1.074)^x}\) is the final answer.