Algebra Applications: Calculating Slope in Tuscany

Tuscany is a region in the northern part of Italy.

Known for its ancient beauty, its old-world charm, and for

its surprisingly steep mountains.

But less well known is how popular cycling is.

The city of Cortona is on a hill and is five hundred

meters above sea level.

It is a city of steep, narrow roads, but that doesn't stop

these cyclists.

Cyclists refer to the steepness of a hill by noting

its percent grade.

For example, a hill with a six percent grade means that for

every one hundred meters you move horizontally you increase

your altitude by six meters.

How does this relate to slope?

In a coordinate system, suppose there are two points

labeled x1, y1, and x2, y2, and suppose there

is a line connecting the two points.

The slope of the line is defined as the rise over the

run, or the ratio of the difference in the y

coordinates over the difference in the

x coordinates.

This is summarized in the slope formula shown here.

So if a hill has a six percent grade then traveling one

hundred meters horizontally corresponds to a six-meter

vertical rise.

We can use these facts to identify two coordinates.

The first coordinate is (0, 0) and the second

coordinate is (100, 6).

Let's use the slope formula to find the slope.

Input the values for y1, y2, x1, and x2.

Let's designate the coordinate 0, 0 as x1-y1, and

let's designate 100, 6 as x2-y2.

Plug in the coordinates into their appropriate slots

and simplify.

The slope of the line connecting the two points

is 0.06

Notice that this is the decimal form of six percent.

This means that the grade of a hill is the same as the slope

written as a percent.

Also, for practical purposes, the grade will never be

greater than a hundred percent since such a grade would be

much too steep.

We can use the y=mx form of a linear equation to

explore different grades.

For example, the graph of y=0.06x is a model of a hill

of grade 6.

Now let's use a TI-Nspire to explore slope.

Turn on the TI-Nspire.

Create a new document.

You may need to save a previous document.

Open a graph window.

Show the background grid by selecting MENU, and under view

selecting SHOW GRID.

Place two points on the grid.

Press MENU, and under POINTS AND LINES select POINT.

Use the nav pad to move the pointer to the origin and

press CLICK to place a point.

Do the same with a point in the first quadrant.

Connect the points of the line by selecting MENU, and under

POINTS AND LINES selecting LINE.

Move the pointer so that it is on top of each point and

press ENTER over each point.

Change the range for the x-axis by moving the pointer

over the axis, holding the SHIFT key, and pressing the

LEFT button.

Choose a max of about a hundred and ten.

Display the coordinates of the points by pressing MENU, and

under ACTION selecting coordinates and equations.

Place the pointer over one of the points and press ENTER

once to see the coordinates, and ENTER again to place the

coordinates in a specific part of the screen.

Move the coordinates to the upper part of the screen.

Repeat the previous two steps with the other point.

Now display the slope of the line by selecting MENU, and

under MEASUREMENT selecting slope.

Move the pointer over the line and click on it.

Then move the label for the slope to the same area of the

screen as you did with the coordinates.

Then press ESCAPE.

Change the coordinates of the two points to 0, 0

and 100, 6.

Move the pointer to highlight the x coordinate of one

of the points.

Press ENTER twice to make the coordinate editable and change

the value.

Repeat with the y coordinate.

Then repeat with the other set of coordinates.

You'll see that the grade is the same as the slope

expressed as a percent.

The five hundred meter rise to Cortona is at a four

percent grade.

How far do these bikers have to travel?

We can use the slope formula to solve this problem.

A four percent grade is the same as the slope of 0.04.

The rise is five hundred meters, so we need to solve

for the run.

The distance the cyclists need to travel is twelve thousand

five hundred meters.

This long distance is made up of a long spiraling road

up to Cortona.

But once you get to the top the view is magnificent.

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