Algebra Applications: Animal Evolution

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Narrator: Animals come in different shapes and sizes.

The diversity of the animal kingdom is due to evolution,

but the mechanism that brings about evolutionary change is

not often clearly understood, nor is it clear what light

math can shed on this process.

Surprisingly, rational functions can provide some

powerful analytical tools for understanding

animal evolution.

This analysis comes from looking at the ratio of

surface area and volume of three-dimensional figures.

All organisms are three-dimensional figures in

space, and all complex shapes can be broken down into simple

3-D figures.

So let's start with some simple examples of surface

area and volume.

This is a cube, one of the simplest

three-dimensional shapes.

Each edge of the cube has length s.

The cube has six sides, and the area of each side is

s-squared.

So the surface area of the cube is six s-squared.

The volume of the cube is s-cubed.

The ratio of the surface area to the volume is six over s.

This is a rational expression and when written in the form f

of x equals six over x it is a rational function.

Now look at this set of eight cubes made up of the

smaller cubes.

The surface area of the larger cube is six times the area of

the face of each side of the larger cube.

Each face has area four s-squared, so the surface area

of the larger cube is six times four s-squared, or

twenty-four s-squared.

The volume is eight times the volume of the small cube so

the total volume is eight s-cubed.

The ratio of surface area to volume is twenty-four s-squared

over eight s-cubed, or three over s.

This is a smaller ratio than the original cube, and you can

see why.

There is less surface area exposed since there are

smaller cubes hidden within the larger cube.

Now look at this configuration of the eight cubes.

Since there are eight small cubes the volume is the same

as before, eight s-cubed.

The surface area, on the other hand, is different.

It is made up of this region, which has area four times

two s-squared.

This region, which is very two times four s-squared and two

times two s-squared, and this region, which has area of four

time two s-squared.

So the total surface area is twenty-eight s-squared.

The ratio of the surface area to volume for this figure is

twenty-eight s-squared divided by eight s-cubed, which equals

three point five over s.

This ratio is larger than the previous configuration because

more surface area is exposed.

Now let's apply these ideas to two examples from the

animal kingdom.

This snake can be thought of as a long cylindrically

shaped object.

This polar bear is a combination of different

cylindrical shapes, but for simplicity let's focus on the

large cylindrical trunk of the polar bear's body.

For any cylinder the surface area is made up of these two

circles which have a combined area of two pi r-squared, and

this rectangular region of length two pi r and width h,

the height of the cylinder.

The total surface area is two-pi r-squared plus

two-pi r h.

The volume of a cylinder is pi r-squared h.

And the ratio of the surface area to volume is

this expression.

Notice that several terms divide out leaving the

expression two times the quantity r plus h divided

by rh.

Both r, the radius of the cylinder, and h, its height,

are variables.

But for this exercise let's assume that the polar bear and

the snake have the same value of h, and for simplicity let h

equal one.

We now have the rational function that we shall analyze

on the TI-Nspire.

Create a new document.

You may need to save a previous document.

Create a graphing window.

At the function entry line input the function two times

the quantity x plus one divided by x.

PRESS ENTER.

This is the graph of a rational function.

Activate the trace feature and use the nav pad to analyze

different coordinates.

As the value of r increases the ratio of surface area to

volume decreases.

This also corresponds to an animal whose width is similar

to that of the polar bear.

On the other hand, as the value of r decreases the ratio

of surface area to volume increases.

This corresponds to an animal whose width is smaller and

which also exposes more surface area.

How does all this relate to animal evolution?

Animals that need to retain heat in their bodies tend to

have a lower surface area to volume ratio, as does the

polar bear.

Animals that need to quickly lose heat have a higher

surface area to volume ratio, as does the snake.

As a result animals that live in cold climates need to

retain body heat and so their surface area to volume ratio

will be lower, meaning that they expose less surface area.

Snakes and other reptiles are cold-blooded animals that tend

to live in hotter climates.

They don't need to retain heat as much and therefore have a

larger surface area to volume ratio so that they can lose

heat quickly.

Animals adapt to their environment and often these

adaptations are affected by their surface area to

volume ratio.

In general, larger animals generate more heat than

smaller animals.

For a polar bear this is important since it lives in a

cold environment.

Fur also helps to keep it warm.

Elephants on the other hand live in a hot climate and

their surface area to volume ratio is low, so they tend to

retain more heat.

As a result elephants have no fur, and they have thin skin

so that they can lose heat more easily.

Elephants also have wrinkled skin, which results in a

larger surface area allowing the elephant to lose heat

more easily.

This also increases the surface area to volume ratio.

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Narrator: The Hubble space telescope is our eye in the

sky, and it has delivered some stunning images.

The inner workings of the Hubble include a network of

mirrors and lenses.

A lens refracts light and focuses it on a point to form

an image.

There at three variables that are of importance with

any lens.

First, there is the focal length of the lens.

This is the point in space where a lens brings an image

into focus.

The focal length is usually described with the letter f.

The second variable is the object distance, in other

words, the distance between the lens and the object you

are looking at through the lens.

The object distance is usually described with the letter o.

The third variable is the image distance.

This is where the refracted light from the lens produces

an image of the object.

The image distance is usually described with letter i.

This equation describes the relationship among f, o,

and i.

For the Hubble, the focal length is a fixed quantity, so

we can rewrite the equation such that i is a function

of o.

This is a rational function that we can graph on the

TI-Nspire.

Turn on the TI-Nspire.

Create a new document.

You may want to save a previous document.

Create a graphs and geometry window.

At the function entry line input the function fifty-seven

point six x divided by the quantity x minus fifty-seven

point six.

Press ENTER.

You'll see part of the curve.

Press MENU and under WINDOW select ZOOM OUT.

Use the nav pad and move the pointer to the center of

the screen.

Press ENTER enough times so that your screen looks

like this.

Notice the horizontal and vertical asymptotes.

We can interpret these asymptotes in terms of the

abilities of the telescope.

For the function, the value of x equals fifty-seven point six

is not allowed since this would result in a zero in

the denominator.

As the value of x approaches fifty-seven point six from the

right the value of y approaches positive infinity.

As the value of x approaches fifty-seven point six from the

left, the value of y approaches negative infinity.

When you move an object in front of a lens the object

distance changes.

When the object distance equals the focal length this

corresponds to the asymptotic value.

At this value the lens does not produce an image.

This is why mathematically an object distance equal to the

focal length is not defined.

Because the Hubble space telescope was built for deep

space observation, then the object distances are vast.

You'll see that as x increases in value the y value, the

image distance, approaches the horizontal asymptote.

To see what the value is as x approaches infinity re-write

the equation in this form.

Now you can see that as x approaches infinity the term

with x in the denominator approaches zero, and the value

for the image distance approaches the focal length.

How well does the Hubble do with nearby objects, like the

moon and other objects in the solar system?

As you can see, as x approaches zero the image

distance increases or decreases in

value dramatically.

The image distance is not as stable as it is with

distant objects.

Here is another problem with nearby objects.

This diagram shows that the angle measure of an object

through a telescope is based on the size of the object and

the distance of the object, as well as the size of the

telescope lens.

The angle is measured in arc seconds where thirty six

hundred arc seconds equal one degree.

The Hubble's resolution is zero point zero five

arc seconds.

This is a very small angle measure and one that benefits

from long distances.

This is why galaxies are clearly visible through

the Hubble.

Galaxies are both huge in size and very distant, which take

advantage of the Hubble's optics.

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