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.)

To address part (c) of your question, we need to determine the age at which an individual can expect to earn the most income using the quadratic function found in part (b). The function given is:

[tex]\[ y = -41.891x^2 + 3987.648x - 49377.617 \][/tex]

This is a quadratic function in the form [tex]$ ax^2 + bx + c $[/tex], where:
- [tex]$ a = -41.891 $[/tex]
- [tex]$ b = 3987.648 $[/tex]
- [tex]$ c = -49377.617 $[/tex]

For any quadratic function of the form [tex]$ ax^2 + bx + c $[/tex], the vertex of the parabola represents the maximum or minimum point, depending on the coefficient [tex]$ a $[/tex]. Since [tex]$ a < 0 $[/tex], the parabola opens downward, and the vertex corresponds to the maximum point.

The x-coordinate of the vertex, which gives us the age at which income is maximized, is found using the formula:

[tex]\[ x = -\frac{b}{2a} \][/tex]

Plugging in the values:

[tex]\[ a = -41.891 \][/tex]
[tex]\[ b = 3987.648 \][/tex]

we get:

[tex]\[ x = -\frac{3987.648}{2 \times -41.891} \][/tex]

Calculating the denominator first:

[tex]\[ 2 \times -41.891 = -83.782 \][/tex]

Now, dividing the numerator by the denominator:

[tex]\[ x = -\frac{3987.648}{-83.782} \][/tex]

[tex]\[ x \approx 47.5955217111074 \][/tex]

Rounding to the nearest tenth:

[tex]\[ x \approx 47.6 \][/tex]

So, at about 47.6 years of age, the individual can expect to earn the most income.