Q. 25 - Precalculus Enhanced with Graphing Utilities

Question

Life Cycle Hypothesis An individual’s income varies with his or her age. The following table shows the median income I of males of different age groups within the United States for $2$$2009.$ For each age group, let the class midpoint represent the independent variable,$x.$ For the class “$65$ years and older,” we will assume that the class midpoint is $69.5.$

(a) Use a graphing utility to draw a scatter diagram of the data. Comment on the type of relationship that may exist between the two variables.

 (b) Use a graphing utility to find the quadratic function of best fit that models the relationship between age and median income. 

(c) Use the function found in part (b) to determine the age at which any individual can expect to earn the most income. 

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

(e) With a graphing utility, graph the quadratic function of best fit on the scatter diagram.

Step-by-Step Solution

Answer

(a) The quadratic model $a<0$ best describes the data.

(b) $I\left(x\right)=-45.122x^2+4301.507x-55376.404.$

(c) An individual can expect the earned most income at the age of $47.1.$

(d) The peak income earned the most at $47143.$

(e) The graph is,

Part (a) Step 1. Given information

The given table is,

We need to use the graphing utility to draw the scatter diagram of the data and comment on the type of relationship that may exist between the two variables.

Step 2. Graphing

The scatter diagram is,

Looking at the scatter diagram we conclude that the quadratic model with$a<0$ best describes the given data.

Part (b) Step 1. Given information

The given diagram is, 

We need to use a graphing utility to determine the quadratic function of best fit.

Step 2. Solving

The quadratic model is of the form.

$y=a{x}^2+bx+c.$

By using the calculation we get,

$a=-45·12196429, b=4301.574821, c=-55376.40353, R^2=.9948537193.$

So, we get,

$y=-45.122x^2+4301.575x-55376.404.$

$I\left(x\right)=-45.122x^2+4301.575x-55376.404.$

Part (c) Step 1. Given information

The given table is,

We need to use the function found in part (b) to determine the age at which any individual can expect to earn the most income.

Step 2. Simplify

To determine the age at which any individual can expect to earn the most income we must determine the $x$-coordinate of the maximum (i.e the vertex). The function is $I\left(x\right)=-45.122x^2+4301.575x-55376.404.$

$x_{max}=-\frac{6}{2a}.$

         $=-\frac{4301.575}{2.\left(-45.122\right)}.$

         $=47.7.$

An individual can expect to earn most income at the age of $47.7.$

Part (d) Step 1. Given information

The given table is, 

We need to use the function in part (b) to predict the peak income earned.

Step 2. Simplify

To determine the peak realized income. We must determine the $y$-coordinate of the maximum (i.e, the vertex).

It $x_v$ is $x$-coordinate of the vertex, then $y_v=f\left(x_v\right)$ is $y$-coordinate of the vertex.

$Y_{MAX}=I\left(47.7\right).$

           $=-45.122·{\left(47.7\right)}^2+4301.575.477-55376.404.$

           $=47143.$

The peak income earned is $47143.$

Part (a) Step 1. Given information

The given table is,

We need to graph the quadratic function of best fit on the scatter diagram with a graphing utility.

Step 2. Graphing

The graph of a quadratic function $I\left(x\right)$