Modelling non-linear Relationships

In many real life experiences we find quantities varying against each other - one quantity, say $y$y, varying in a predictable way simply because another related quantity, say $x$x, is varying. This might be the height of a cricket ball varying because of a changing angle of projection. It might be the the distance to the horizon varying as the observer changes her distance from the earth. It might be the area of a rectangle varying as its dimensions change.

Whenever this occurs, not only might it be possible to formulate why the variation happens, but also the way it happens. 

Linear variation is variation in $y$y that occurs as a fixed multiple $m$m of the variation in $x$x. It gets its name from the type of graph that is produced when $y$y is plotted against $x$x - a straight line of gradient $m$m.

But there are many other types of variation, and these are often characterised as non-linear. We will explore two instances of non-linear variation here.

Example 1: The Area of an enclosure. 

A property owner, Anne, decides to set up a rectangular grassed enclosure for her horse. She has 200 metres of fencing, and in an effort to increase the grazing area, has used a straight section of a creek as one of the sides (see diagram below).

She originally thought that no matter what the dimensions of the rectangular enclosure she erects, the feeding area for the horse wouldn't change. However, after doing some mathematics, she realised that the area varies as the dimensions vary. 

Her mathematical logic was clever. Setting the two shorter sides as the variable length $x$x metres, Anne determines that the longer side must be $200-2x$200−2x. The area then can be expressed as a function of the shorter sides, given as $A=l\times b=x\left(200-2x\right)$A=l×b=x(200−2x).

Expanding this last expression we have $A=200x-2x^2$A=200x−2x2. Anne could try various values of $x$x to see how the area would vary, but instead decides to graph the area as a function of $x$x. It's an inverted parabola with roots of $0$0 and $100$100.

She immediately recognises that the area can be maximised if $x=50$x=50 is chosen. This means, by substitution, that $A=200\left(50\right)-2\left(50\right)^2=5000$A=200(50)−2(50)2=5000 square metres of grazing is possible. 

 

Example 2: Kepler's Third Law

Johannes Kepler (1571 - 1630) discovered an important relationship between a planet's mean distance from the Sun and its period (the time taken to get around the Sun).

If we call the distance the Earth is from the Sun $1$1 astronomical unit ($1$1AU for short) then we can write down each planet's distance in AUs and each planet's period (in "Earth" years). The tables shows this data broken up into the inner and outer planets of our solar system.

Table 1: Inner Planets	Mercury	Venus	Earth	Mars
Distance (R)	0.387	0.723	1	1.523
Period (T)	0.241	0.615	1	1.881

 

Table 2: Outer Planets	Jupiter	Saturn	Uranus	Neptune
Distance (R)	5.203	9.537	19.191	30.069
Period (T)	11.9	29.5	84.0	164.8

Kepler was able to prove, given certain assumptions, that the relationship between the period $T$T in Earth years and the distance $R$R was approximately given by:

$T=R\sqrt{R}$T=R√R

Looking at the inner planets first, Kepler's formula determines the periods as: 

$T_{Me}$TMe​	$=$=	$0.387\times\sqrt{0.387}=0.241$0.387×√0.387=0.241
$T_{Ve}$TVe​	$=$=	$0.723\times\sqrt{0.723}=0.615$0.723×√0.723=0.615
$T_{Ea}$TEa​	$=$=	$1\times\sqrt{1}=1$1×√1=1
$T_{Ma}$TMa​	$=$=	$1.523\times\sqrt{1.523}=1.880$1.523×√1.523=1.880

We could do the same for the outer planets. For example, Kepler's Law for the planet Neptune shows that $T_{Ne}=30.069\times\sqrt{30.069}=164.884$TNe​=30.069×√30.069=164.884, which is remarkably close for such a far off planet. Perhaps you could check the other 3 planets yourself.

We can sketch the function $T=R\sqrt{R}$T=R√R across the restricted domain $0\le R\le35$0≤R≤35 to understand the non-linearity of the relationship. Below is such a sketch showing the approximate relative positions of Jupiter, Saturn, Uranus and Neptune. What a special natural relationship this is.

 

Further Examples

Question 1

Question 2

Question 3