Chapter 5.8, Problem 34PPSE

Solution

To determine : The curve that seems most likely to model the gross domestic product over the years.

The quartic seems most likely to model the gross domestic product over the years.

Given information :

The table given below shows the United States gross domestic product for selected years.

Year	GDP (Billion $)
$\begin{array}{l}1960\\ 1970\\ 1980\\ 1990\\ 2000\end{array}$	$\begin{array}{l}526.4\\ 1038.5\\ 2789.5\\ 5803.1\\ 9817.0\end{array}$

Explanation :

Construct quartic and cubic model for this data and take 1960 as the base year, where $x=0$ .

The quartic model will be $y=a{x}^4+b{x}^3+c{x}^2+dx+e$ .

The values are $a=-0.0012,b=0.075,c=4.765,d=-2.795,e=526.4$ and $r^2$ is $1$ .

So, the quartic model will be:

  $y=-0.0012x^4+0.075x^3+4.765x^2-2.795x+526.4$

The cubic model will be $y=a{x}^3+b{x}^2+cx+d$ .

The values are - $a=-0.20,b=7.10,c=-19.82,d=530.49$ and $r^2$ is $0.9999$ .

So, the cubic model will be:

  $y=-0.20x^3+7.10x^2-19.82x+530.49$

The quartic seems most likely to model the gross domestic product over the years as the $r^2$ is $1$ .