Chapter 3.4, Problem 25AYU

(a)

To determine

To draw: The scatter diagram of the table.

Answer to Problem 25AYU

Explanation of Solution

Given: An individual’s income varies with his or her age. The below table shows the median income I of males of different age groups within the United States for 2006. For each age group, let the class midpoint represent the independent variable x . For the class 65-years and older.

  $\begin{array}{ccc}\text{Age}& \text{Class Midpoint,}x& \text{Median Income, }I\\ 15-24\text{ years}& 19.5& \$10,964\\ 25-34\text{ years}& 29.5& \$32,131\\ 35-44\text{ years}& 39.5& \$42,637\\ 45-54\text{ years}& 49.5& \$45,693\\ 55-64\text{ years}& 59.5& \$41,477\\ 65\text{ years and older}& 69.5& \$23,500\end{array}$

Using graphing utility to draw scatter plot.

(b)

To determine

To find: The quadratic function of best fit that models.

Answer to Problem 25AYU

  $y=-0.0455x^2+4.3144x-55.962$

Explanation of Solution

Given: An individual’s income varies with his or her age. The below table shows the median income I of males of different age groups within the United States for 2006. For each age group, let the class midpoint represent the independent variable x . For the class 65-years and older.

  $\begin{array}{ccc}\text{Age}& \text{Class Midpoint,}x& \text{Median Income, }I\\ 15-24\text{ years}& 19.5& \$10,964\\ 25-34\text{ years}& 29.5& \$32,131\\ 35-44\text{ years}& 39.5& \$42,637\\ 45-54\text{ years}& 49.5& \$45,693\\ 55-64\text{ years}& 59.5& \$41,477\\ 65\text{ years and older}& 69.5& \$23,500\end{array}$

Using graphing utility, the equation of quadratic of best fit.

  $y=-0.0455x^2+4.3144x-55.962$

(c)

To determine

To find: The age at which an individual earned the most income from the model $y=-0.0455x^2+4.3144x-55.962$ .

Answer to Problem 25AYU

  $x=47\text{ years}$

Explanation of Solution

Given: An individual’s income varies with his or her age. The below table shows the median income I of males of different age groups within the United States for 2006. For each age group, let the class midpoint represent the independent variable x . For the class 65-years and older.

  $\begin{array}{ccc}\text{Age}& \text{Class Midpoint,}x& \text{Median Income, }I\\ 15-24\text{ years}& 19.5& \$10,964\\ 25-34\text{ years}& 29.5& \$32,131\\ 35-44\text{ years}& 39.5& \$42,637\\ 45-54\text{ years}& 49.5& \$45,693\\ 55-64\text{ years}& 59.5& \$41,477\\ 65\text{ years and older}& 69.5& \$23,500\end{array}$

Using graphing utility, the equation of quadratic best fit.

  $I=-0.0455x^2+4.3144x-55.962$

This is quadratic equation. At which age earn most income.

  $x=47$

(d)

To determine

To find: The maximum income earn over the age.

Answer to Problem 25AYU

  $I_{\max }=\$46,313$

Explanation of Solution

Given: An individual’s income varies with his or her age. The below table shows the median income I of males of different age groups within the United States for 2006. For each age group, let the class midpoint represent the independent variable x . For the class 65-years and older.

  $\begin{array}{ccc}\text{Age}& \text{Class Midpoint,}x& \text{Median Income, }I\\ 15-24\text{ years}& 19.5& \$10,964\\ 25-34\text{ years}& 29.5& \$32,131\\ 35-44\text{ years}& 39.5& \$42,637\\ 45-54\text{ years}& 49.5& \$45,693\\ 55-64\text{ years}& 59.5& \$41,477\\ 65\text{ years and older}& 69.5& \$23,500\end{array}$

Using graphing utility, the equation of quadratic best fit.

  $I=-0.0455x^2+4.3144x-55.962$

This is quadratic equation. At which age earn most income.

  $x=47$

Therefore, $I_{\max }=\$46,313$

(e)

To determine

To find: The domain of function.

Answer to Problem 25AYU

Explanation of Solution

Given: An individual’s income varies with his or her age. The below table shows the median income I of males of different age groups within the United States for 2006. For each age group, let the class midpoint represent the independent variable x . For the class 65-years and older.

  $\begin{array}{ccc}\text{Age}& \text{Class Midpoint,}x& \text{Median Income, }I\\ 15-24\text{ years}& 19.5& \$10,964\\ 25-34\text{ years}& 29.5& \$32,131\\ 35-44\text{ years}& 39.5& \$42,637\\ 45-54\text{ years}& 49.5& \$45,693\\ 55-64\text{ years}& 59.5& \$41,477\\ 65\text{ years and older}& 69.5& \$23,500\end{array}$

Using graphing utility, the quadratic best fit equation.