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An individual's income varies with age. The table shows the median income [tex]\( I \)[/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]\( x \)[/tex]. For the class "65 years and older," assume that the class midpoint is 69.5.

Complete parts (a) through (e).

[tex]\[
\begin{tabular}{|lrc|}
\hline
\multicolumn{1}{|c}{ Age } & \begin{tabular}{c}
Class \\
Midpoint, \\
X
\end{tabular} & \begin{tabular}{c}
Median \\
Income, I
\end{tabular} \\
\hline
15-24 years & 19.5 & \$12,965 \\
25-34 years & 29.5 & \$31,130 \\
35-44 years & 39.5 & \$42,637 \\
45-54 years & 49.5 & \$44,692 \\
55-64 years & 59.5 & \$41,477 \\
65+ years & 69.5 & \\
\hline
\end{tabular}
\][/tex]

The quadratic function of best fit is
[tex]\[ y = -41.891x^2 + 3987.648x - 49377.617 \][/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 47.6 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.)

(d) Use the function in part (b) to predict the peak income earned.

The predicted peak income is about [tex]\(\$ \square\)[/tex].
(Round to the nearest dollar as needed.)



Answer :

GinnyAnswer
To predict the peak income using the quadratic function of best fit, we follow these steps:

1. Understand the Given Quadratic Function:
The quadratic function that models the income in relation to age is given by:
[tex]\[ y = -41.891 x^2 + 3987.648 x - 49377.617 \][/tex]
where [tex]\( y \)[/tex] represents the income and [tex]\( x \)[/tex] represents the age.

2. Determine the Age at which Peak Income Occurs:
From part (c), the age at which an individual can expect to earn the most income is 47.6 years. Therefore, [tex]\( x = 47.6 \)[/tex] is used to find the peak income.

3. Predict the Peak Income:
To find the peak income, substitute [tex]\( x = 47.6 \)[/tex] into the quadratic function:
[tex]\[ y = -41.891 (47.6)^2 + 3987.648 (47.6) - 49377.617 \][/tex]
Based on the given solution, calculating this expression will yield:
[tex]\[ y \approx 45519.475640000004 \][/tex]
Rounding this to the nearest dollar, the predicted peak income is:
[tex]\[ \$45,519 \][/tex]

Thus, the predicted peak income is about \$45,519.