An individual's income varies with age. The table shows the median income [tex][tex]$I$[/tex][/tex] of individuals of different age groups within the United States for a certain year. For each age group, let the class midpoint represent the independent variable [tex][tex]$x$[/tex][/tex]. For the class "65 years and older," assume that the class midpoint is 69.5.
Complete parts (a) through (e).
\begin{tabular}{|l|cc|}
\hline \multicolumn{1}{|c}{ Age } & \begin{tabular}{c}
Class \\
Midpoint,
\end{tabular} & \begin{tabular}{c}
Median \\
Income, I
\end{tabular} \\
\hline 15-24 years & 19.5 & [tex][tex]$\$[/tex]12,965[tex]$[/tex] \\
\hline 25-34 years & 29.5 & [tex]$[/tex]\[tex]$31,130$[/tex][/tex] \\
\hline 35-44 years & 39.5 & [tex][tex]$\$[/tex]42,637[tex]$[/tex] \\
\hline 45-54 years & 49.5 & [tex]$[/tex]\[tex]$44,692$[/tex][/tex] \\
\hline 55-64 years & 59.5 & [tex][tex]$\$[/tex]41,477[tex]$[/tex] \\
\hline 65 years and older & 69.5 & [tex]$[/tex]\[tex]$24,502$[/tex][/tex] \\
\hline
\end{tabular}
(a) Determine the relation between age and median income.
(b) Use a graphing utility to find the quadratic function of best fit that models the relation between age and median income.
The quadratic function of best fit is [tex][tex]$y = -41.891x^2 + 3987.648x - 49377.617$[/tex][/tex].
(Type integers or decimals rounded to three decimal places as needed.)
(c) Use the function found in part (b) to determine the age at which an individual can expect to earn the most income.
At about [tex]\square[/tex] years of age, the individual can expect to earn the most income.
(Do not round until the final answer. Then round to the nearest tenth as needed.)