Q. 29 - Precalculus Enhanced with Graphing Utilities

Question

Which Model? The following data represent the birth rate (births per $1000$population) for women whose age is $x,$ in $2007.$ 

(a) Using a graphing utility, draw a scatter diagram of the data treating age as the independent variable. What type of relationship appears to exist between age and birth rate?

 (b) Based on your response to part (a), find either a linear or quadratic model that describes the relation between age and birth rate.

 (c) Use your model to predict the birth rate of $35$-year-old women. 

Step-by-Step Solution

Answer

(a) The scatter diagram is,

(b) $B\left(a\right)=-0.495a^2+26.928a-256.398.$

(c) Birth rate is $79.7.$

Part (a) Step 1. Given introduction

The given table is 

We need to draw a scatter diagram of the data treating age as the independent variable and determine the type of relationship that appears to exist between age and birth rate.

Step 2. Graphing

The scatter diagram of the data is,

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

Part (b) Step 1. Given introduction

The given table is,

Based on the result to part (a) we need to determine either a linear or quadratic model to describe the relationship between age and birth rate.

Step 2. Simplify

The quadratic model is of the form

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

Using the calculator and the values given in the table we determine the value of constant $a,b, and c.$

So, $a=-0.494666667, b=26.92771429, c=-256.3981905.$

Therefore, $y=-0.495x^2+26.928x-256.398.$

                   So, $B\left(a\right)=-0.495a^2+26.928a-256.398.$

Part (c) Step 1. Given introduction

The given table is, 

We need to use the model to predict the birth rate of $35$ years old woman.

Step 2. Simplify

$B\left(a\right)=-0.495a^2+26.928a-256.398.$

If $a=35.$

$B\left(35\right)=-0.495{\left(35\right)}^2+26.928\left(35\right)-256.398.$

            $=79.7.$