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?

No.

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.