Algebra Applications: Rational Functions
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Title: Algebra Applicatons: Rational Functions
Title: Submarines
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Narrator: Submarines are sophisticated vehicles that
need one thing more than anything else to remain
underwater in one piece, strength.
There are massive forces that push against the submarine and
the deeper the submarine goes into the ocean the greater the
force of water pressure.
These forces are so great that depending on the structure of
the submarine there are depths beyond which it cannot go
without damaging it.
How does water pressure damage a submarine?
Simply put, when the outside forces of water pressure
exceed the structural integrity of the submarine the
vessel implodes.
No one can survive such a devastating fate.
This chart shows the increase in water pressure as the depth
under water increases.
The unit of pressure is an atmosphere.
The value of one atmosphere is what is felt at sea level.
Let's use the TI-Nspire to analyze this data set.
Turn on the TI-Nspire and create a new document.
You may need to save your work from a previous document.
Create a spreadsheet.
Use the nav pad to move to the top of column A.
Input x as the column heading.
Move the cursor to the top of column B and input y.
Move to cell A1 and begin inputting the data in
the table.
Pause the video to input the data.
After you have input the data, create a statistics window.
Use the nav pad to highlight the x-axis.
Select the x variable.
Repeat with the vertical axis and select the y variable.
Your screen should look like this.
Notice that there is a linear relationship between depth
and pressure.
Let's attach an empty plastic bottle to the submarine, which
has descended.
As the depth increases the pressure increases.
The submarine can withstand the increased pressure, but
not the plastic bottle.
What happens to the bottle?
After a certain depth the bottle collapses.
The volume in the bottle literally decreases.
So there is an inverse relationship between pressure
and volume.
As the water pressure increases the volume of
air decreases.
This can be written as an equation of this form.
The letter c stands for a constant.
For simplicity let's make c equal to one and use the
letters p and v for pressure and volume.
The equation becomes p equals one over v, and this is an
example of a rational function.
Let's take a look at the graph of this function on
the Nspire.
Continuing with the previous document, create a
graphing window.
We will use y for p and x for v to graph the function.
So at the function entry line type in one divided by x and
PRESS ENTER.
Restrict the graph window to the first quadrant.
Now activate the trace feature to analyze the coordinates.
Use the nav pad to see how the coordinates change.
The x-axis represents pressure, in this case the
independent variable, and the y-axis represents the volume,
in this case the dependent variable.
Watch what happens as the value of p increases.
The value v decreases.
What this means is that as the pressure increases the volume
decreases, and this represents the submarine descending.
But no matter how large the value of p gets v never
equals zero.
For this graph the x-axis is called an asymptote.
Now watch what happens as the value of p decreases.
The value of v increases.
This means that as the pressure decreases the
volume increases.
This represents the submarine rising to the surface.
But no matter how small the value of p gets it doesn't
equal zero.
This makes sense because by definition a rational function
of the form f of x over g of x cannot have g of x
equal to zero.
This makes the y-axis another asymptote of the graph.
Now, when a submarine is descending the water pressure
surrounding the submarine increases, but the volume of
the submarine, its size, doesn't change.
Doesn't this contradict what the rational function
graph shows?
The submarine is built to withstand the
increased pressure.
It is built to maintain its volume.
Without this hard structure the sides of the vessel would
collapse at greater ocean depths.
One other observation.
As you can see from the graph, as the pressure decreases the
volume increases.
This is an issue that scuba divers have to deal with.
A scuba diver relies on pressurized air in order to
breathe under water.
This pressurized air includes a larger concentration of
nitrogen than is found at sea level.
If a scuba diver surfaces too quickly the volume of nitrogen
gas in the bloodstream expands causing a physical reaction
known as the bends.
A severe case of the bends can cause death.
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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
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,
For the Hubble, the focal length is a fixed quantity, so
we can rewrite the equation such that i is a function
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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