Chapter 5.8, Problem 28PPSE

Solution

To calculate : The carbon-dioxide in Earth’s atmosphere in the year 2022.

The carbon-dioxide in Earth’s atmosphere in the year 2022 is $569.98\text{ ppm}$ .

Given information :

The data is:

Year	CO2 in atmosphere (ppm)
$\begin{array}{l}1968\\ 1983\\ 1998\\ 2003\\ 2008\end{array}$	$\begin{array}{l}324.14\\ 343.91\\ 367.68\\ 376.68\\ 385.60\end{array}$

Formula used:

  $\left(n+1\right)$ Principle: For any set of $n+1$ points in the coordinate plane that pass the vertical line test, there is a unique polynomial of degree at most $n$ that fits the point perfectly.

Calculation :

Let $x$ represent the years after 1900 and $y$ represent the CO₂ in atmosphere.

$x$	$y$
$\begin{array}{l}68\\ 83\\ 98\\ 103\\ 108\end{array}$	$\begin{array}{l}324.14\\ 343.91\\ 367.68\\ 376.68\\ 385.60\end{array}$

Here, there are five set of points. From the $\left(n+1\right)$ principle, there is a unique polynomial of degree at most $4$ that fits the points perfectly.

To determine the model that fits the data calculate linear, quadratic, cubic and quartic model for this data.

For this find the linear, quadratic, cubic and quartic regression to get the models using the calculator.

The linear model will be:

  $y=ax+b$

Here, $a=1.54,b=218.20$ .

So, the linear function will be: $y=1.54x+218.20$ and $r^2=0.9965$ .

The quadratic model will be:

  $y=a{x}^2+bx+c$

Here, $a=0.009,b=0.015,c=283.03$ .

So, the quadratic function will be: $y=0.009x^2+0.015x+283.03$ and $r^2=0.99997$ .

The cubic model will be:

  $y=a{x}^3+b{x}^3+cx+d$

Here, $a=-3.77\times {10}^{-5},b=0.02,c=-0.85,d=307.60$ .

So, the cubic function will be: $y=\left(-3.77\times {10}^{-5}\right)x^3+0.02x^2-0.85x+307.60$ and $r^2=0.99998$ .

The quartic model will be:

  $y=a{x}^4+b{x}^3+c{x}^3+dx+e$

Here, $a=-1.37\times {10}^{-5},b=0.005,c=-0.64,d=37.52,e=-525.94$ .

So, the quartic function will be: $y=\left(-1.37\times {10}^{-5}\right)x^4+0.005x^3-0.64x^2+37.52x-525.94$ and $r^2=1$ .

By comparing the values of $r^2$ , the quartic model best fits the data.

Therefore, the model best fits the data is $y=\left(-1.37\times {10}^{-5}\right)x^4+0.005x^3-0.64x^2+37.52x-525.94$ .

To find the carbon-dioxide in Earth’s atmosphere in the year 2022, substitute $x=2022-1900=122$ in the quartic model.

  $\begin{array}{c}y=\left(-1.37\times {10}^{-5}\right){\left(122\right)}^4+0.005{\left(122\right)}^3-0.64{\left(122\right)}^2+37.52\left(122\right)-525.94\\ =-3035+9079.24-9525.76+4577.44-525.94\\ =569.98\text{ ppm}\end{array}$