Algebra Applications: Inequalities
Algebra Applications: Inequalities
[Music]
[Music]
[Music]
Title: Inequalities
Title: Hybrid Cars
Narrator: We live in a time when a car's interior is more
important than its exterior.
With more and more people driving worldwide the
increased demand for fossil fuels has made
gasoline expensive.
So car engines that use less gasoline for every mile
traveled are becoming more and more popular and more and
more necessary.
The car on the left is a traditional gasoline
powered car.
Its internal combustion engine burns gasoline.
As the gasoline is burned the pistons in the engine convert
chemical energy into the energy of motion and this is
what powers the car.
But this power comes at a cost.
When a car is at a stop sign or a red light the engine is
still burning gasoline.
The Pistons are still moving but the car is stationary.
This is a waste of energy but for many years gasoline was
cheap and the extra energy, all things being equal, was
not a big loss.
The car on the right is a hybrid.
It still has a gasoline powered engine but it's a
smaller engine that uses less gasoline.
What makes the hybrid energy efficient is the use of a
battery system that powers an electric motor.
During times when the car is idle at stop signs, red
lights, and other similar places, the electric
system takes over.
The car's engine temporarily shuts off allowing the car to
preserve its gasoline.
As a result a hybrid car travels more miles per gallon
or MPG than a gasoline powered car.
A car's mpg depends on two factors; how much driving is
done in the city where there are many stops and starts, and
how much is done on the highway where there are far
fewer stops.
On average a car has a low MPG in city traffic and the higher
MPG on the highway.
This is true of both gasoline engines and hybrid engines.
Let's compare these differences.
This car gets 12 miles per gallon in the city and has 20
mpg on the highway.
This hybrid gets 21 mpg in the city and 45 mpg on
the highway.
The owner of the gasoline powered car and the owner of
the hybrid each drive up to 20,000 miles per year through
a combination of city and highway traffic.
Let x represent the number of gallons of gasoline used in
city traffic and y represent the number of gallons used in
highway traffic.
There are two linear inequalities that result from
the situation; 12 x + 20 y is less than or equal to 20,000,
and 21 x + 45 y is less than or equal to 20,000 .
Before analyzing these inequalities let's look at the
corresponding graphs of the equations to see what we
can learn.
The equations can be rewritten to show y is a function of x
as shown.
Lets take a look at the graphs of these equations on the
TI-Nspire.
Turn on the TI-Nspire, create a new document.
You may need to save a previous document.
Open a graph window.
The cursor is on function F1.
Input the first function.
Press CONTROL and the DIVISION KEY to create a
frACTION placeholder.
Complete inputting the entire function then press the DOWN
ARROW to go to the F2 function entry line.
At the F2 entry line, input the second function.
Again, press CONTROL and the DIVISION KEY to create a
frACTION placeholder.
Press ENTER to graph both functions.
Notice that the graph window is empty.
The graphs are there but are not visible.
Change to WINDOW SETTINGS by pressing MENU and under
WINDOWS ZOOM select the ZOOM FIT option.
Now you can see the graphs.
Now that you can see the graphs you can further refine
the WINDOW SETTINGS by pressing MENU and under
WINDOWS ZOOM selecting WINDOW SETTINGS.
Change x Min and y Min to -200.
and x Max and y max to 1600.
Press TAB to move from one field to another.
Press ENTER after you have tabbed to the OK button.
These graphs give the amount of gasoline used for each car
to reach the 20,000 mile mark.
The x values are the amount of fuel use for city driving and
the y values are for the amount of fuel used for
highway driving.
Let's look at some examples but first let's clear up the
screen a bit.
Use the nav pad to hover over the text of the equations.
Press and hold the CLICK key until the open hand becomes a
closed hand.
Use the nav pad to drag the label to the upper part
of the screen.
Repeat with the other equation label.
Now, press MENU and under TRACE select GRAPH TRACE.
Use the nav pad to slide the point through to the right to
see different values for x and Y.
Input 100 to see what the value is.
Press the DOWN ARROW to see the corresponding values for
the other function, the one representing the hybrid car.
This table summarizes the differences in the amount of
fuel needed.
In the example shown the gasoline engine needs twice as
much fuel as the hybrid engine, or about 500
more gallons.
In the summer of 2008 the price of gasoline reached a
price of four dollars per gallon which means that the
driver of the car with the gas engine could pay up to $2000
more in fuel costs than the driver of the hybrid.
But the linear graph shows that there are many
combinations of city and highway mileage.
How can we compare different scenarios for the gasoline
engine and the hybrid engine?
For each graph the sum of the x and y values is the total
amount of gasoline needed to get to 20,000 miles.
Lets track how these x and y values change.
We'll add a point to each line and track its coordinates.
Press MENU and under POINTS AND LINES select POINT.
Use the nav pad to move the pointer above the first graph.
Press ENTER to place the point then press ESCAPE.
Now assign variables to the x and y coordinates.
Use the nav pad to hover over the x coordinate.
Press the variable key labeled var and select STORE.
Assign this value to the variable A.
Press ENTER.
Repeat with the y coordinate.
Assign its values to variable B.
Press ENTER.
Press the TAB key to go to the function entry line.
Input the function A + B and press ENTER.
This horizontal line tracks the total amount of gas used
for all the possible scenarios where the gasoline powered
engine travels 20,000 miles.
Use the nav pad to hover over the point.
Hold the CLICK key to highlight it, now use the LEFT
and RIGHT ARROWS to move the point.
Notice how the horizontal line moves up and down.
What does this mean?
As you move to the right the amount of city driving
increases and since this has a lower MPG you will need more
fuel to reach 20,000 miles.
As you move to the left the amount and city driving
decreases which means the amount of highway
driving increases.
With its higher MPG the increased highway driving
means that you need less fuel to reach 20,000 miles.
Now lets compare the situation with the hybrid car.
Press MENU and under POINTS AND LINES select POINT.
Use the nav pad to move to the second graph.
Add the point by pressing ENTER then press ESCAPE.
Hover over the x coordinate.
Store this value into the variable C and press ENTER.
Hover over the y coordinate, store this value into the
variable D and press ENTER.
Press TAB to go to the function entry line.
Input the function C + D and press ENTER.
Use the nav pad to hover over the second point to
highlight it.
Slide the point across the line to see the graph of the
total amount of gasoline needed change.
There is a scenario where the amount of gasoline needed for
the hybrid is almost the amount needed for the
gasoline engine.
This is a case where the gasoline engine is used only
for highway driving and the hybrid car is used only for
city driving.
In fact a family that has two cars, one a gasoline engine
and the other a hybrid, would do well with this strategy but
most drivers combine highway and city driving and in all
of these scenarios the hybrid car uses much less fuel.
In fact the driver of the car with gasoline engine would
need to think of ways of cutting back on fuel in order
to save on cost.
Suppose this driver has decided to keep his or her
gasoline engine car for another two years before
switching to a hybrid car.
The driver doesn't want to drive more than 20,000 miles.
Let's examine some scenarios involving the inequality.
Press CONTROL + G to bring back the function entry line.
Use the UP ARROW to bring up the F1 function.
Use the LEFT ARROW key and the CLEAR button to delete
the=sign.
Replace it with the ≤ symbol and
press ENTER.
This inequality makes many more scenarios available for
the driver of the gasoline engine car.
Let's investigate these scenarios by adding a point.
Press MENU and under POINTS AND LINES select POINT.
Use the nav pad to place the point in the shaded region
displaying the coordinates of the point.
Press MENU and under ACTION select COORDINATES
AND EQUATIONS.
Use the nav pad to hover over the point and press ENTER to
see the coordinates, then press ESCAPE.
Link these coordinates to two new variables.
Use the nav pad to hover over the x coordinate, press the
var key, and store the value of the x coordinate in the
variable E.
Press ENTER.
Do the same for the y coordinate.
Store the value of the y coordinate in variable F.
Press ENTER.
Press CONTROL + G to bring back the function entry line.
Input the function E + F and press ENTER.
You can clean up the screen by pressing MENU and under
ACTION selecting HIDE/SHOW.
Hide equations, graphs, and labels by clicking on the icon
that looks like an eye to deselect it.
After you are done selecting items to hide press ESCAPE.
Try to get your screen to look like this.
Now you are ready to explore different scenarios.
Use the nav pad to hover over the point that represents the
amount of gasoline used for less than 20,000 miles.
Press and hold the CLICK key until the open hand turns into
a closed hand.
Use the nav pad to move the point to different positions.
Notice how the graph of F5 changes.
There a number of scenarios that allow this driver to use
nearly as much gasoline as the hybrid.
But what is the impact on the amount of mileage traveled?
We can create a formula to track the mileage.
Press MENU and under ACTIONS select TEXT.
Use the nav pad to move to the top of the screen and input
the following text label.
Press ENTER.
Continuing with the TEXT TOOL input this formula.
This is a formula for calculating the total number
of miles traveled for the gasoline powered car.
Link this formula to the coordinate of our point in the
shaded region.
Press MENU and under ACTION select CALCULATE.
Use the nav pad to hover over the formula.
Press ENTER.
Use the nav pad to hover over the x coordinate of the point.
Press ENTER.
Repeat for the y coordinate.
Now start moving the point to different locations in the
shaded region.
Notice how the total number of gallons changes as does the
total number of miles traveled.
Since the point is in the shaded region the driver will
travel less than 20,000 miles but this model allows you to
find scenarios in which the amount of gasoline used is
comparable to that of the hybrid car while the total
number of miles traveled is not too much lower
than 20,000.
As we transition from gasoline powered vehicles to hybrids
and other forms of transportation we can still
take advantage of reliable technologies through an
analysis of linear inequalities.
[Music]
Venice is slowly sinking into the sea but this is a city
that is always embraced the water.
Because the city is at sea level and the water is always
nearby Venice floods very easily.
For centuries the city has endured constant floods but
now as the city itself begins to sink and the sea level
itself is expected to rise over the next century Venice
finds itself in a struggle for its own survival.
As you can see from above the city is surrounded by water
and its main road is a large canal.
There is a natural barrier that may provide a
possible solution.
One plan under consideration is to build a movable
floodgate that can keep enough water at bay during heavy
storms to keep the city from flooding.
This incredible engineering feet could help keep Venice
dry but this floodgate needs to be activated only when the
sea level reaches a critical level during heavy rains.
But what is the mechanism that would make such a
thing possible?
This complex floodgate would depend on a simple inequality.
If we let x represent the water level, and let A
represent the critical value beyond which the flood gate is
activated then the inequality x is greater than or equal to
A determines when the floodgate is activated.
Let's use the TI-Nspire to create a model of
this inequality.
Turn on the TI-Nspire.
Created a new document.
You may need to save a previous document.
Open a list and spreadsheet window.
Lets define a random variable that will be in the varying
depth of the water around Venice.
We'll let this number vary from 40 to 60 and put this
formula into cell A1 and press ENTER.
This will generate a random number from 40 to 60.
Press CONTROL and R several times to test the function by
generating several random numbers within the set range.
We now need to create a function that tests different
iterations of this random value to see if it has passed
a threshold level.
Press the DOC key and under insert select the
program editor.
At the dialog box create a function called Depth Check
as shown.
Press TAB, the DOWN ARROW twice, and ENTER to
select function.
Press TAB twice to select the OK button and press
Notice that this creates a new tabbed window.
The spreadsheet is still there but in a separate window.
The depth check function will take an input value and
evaluate it.
So input the letter A as shown and press the
DOWN ARROW.
A function always returns a value or an expression.
So we defined the function so that it evaluates input value
A and it returns a message to either raise the
floodgate or not.
For our floodgate suppose that when A is greater than or
equal to 55 the gate needs to go up.
Otherwise the gate stays down.
Input the if statement with the inequality as shown.
Be sure to use the SPACE key between the if and the
variable A and elsewhere as shown.
Once the function is defined press CONTROL and B
to save it.
Use this command any time you make changes to the function.
The function will evaluate the input and will determine if
the floodgate comes up or stays down.
A function like this can be used in a computer program
that controls the floodgate mechanism.
Let's test this function with the spreadsheet value.
Activate the spreadsheet window.
Make sure that cell A2 is the active cell.
Input the newly defined function depth check and have
it evaluate the value in cell A1 where the random var
is generated.
Press ENTER.
Press CONTROL and R several times to test
different values.
You'll notice that for the most part the gate stays down
and comes up only on rare occasions.
This is an accurate model of how the floodgate system would
work, only in the infrequent cases were severe storm is
expected to cause flooding.
Venice is a beautiful city with a glorious history.
Preserving this city may ultimately rely on the inner
workings of an inequality.
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