Eiffel Tower

Parabola Project

Katie S.



Parabola Project: The Eiffel Tower

The Eiffel Tower was built in 1889. Its architect was the talented engineer, Gustave Eiffel. He worked for years on air resistance experiments and mathematical equations that could be applied to its building, because he was extremely worried that due to the tower’s size, mass, and shape, it would topple over not long after being built. It was comprised of eighteen thousand different parts, which were built in a workshop and then assembled on site. It was the tallest building in the world at 312 m, but was topped several years later by the Chrysler Building, which was built 319 m tall. Today it is 324 m tall with the addition of radio antennae. Its total weight is ten thousand and ten tons. Crazy things happen there every so often, like guys driving up the tower in mountain bikes or a diving pool being installed at the foot of the tower.
EiffelTowerLit-thumb-400x533-1676.png

The first layer of the tower is comprised of four supports that form four base arches. Staircases are within each of these supports. People can climb up the tower by the stairs or by a lift goes up and down, stopping at each platform of the tower. There is a viewing level just above the base arches. Higher up, there is another viewing level. You can also go to the top of the Eiffel Tower. Gift shops and cafes are kind of scattered throughout the various levels. The tower is maintained and checked regularly. Every evening, it will flash for ten minutes every hour until midnight. I visited the Eiffel Tower once and saw all of this in person. It kind of freaked me out because I really don't like high places.

There are numerous parabolas in and around the Eiffel Tower, but the one I am researching is one of the base arches of the tower.
To start, here is my graph of this parabola.


Eiffel_Tower.jpg
In this graph, the y axis represents the axis of symmetry and the height of the parabola/base arch. The x axis represents the ground beneath the Eiffel tower. The vertex of my graph is (0,57), the maximum. It represents the base arch's full height. The vertex is also the y-intercept of the parabola. I found it simply by finding out the height of the base arch, 57 meters, and then deciding to use the y axis as the axis of symmetry, so that the x value would be 0. This parabola clearly has two x-intercepts, (-33,0) and (33,0). The maximum width of the base arch is 66 meters, at ground level. Because I had already designated the y axis as the axis of symmetry, and designated the x axis as the ground level, all I had to do to find the x-intercepts was to split the maximum width in half, hence -33 and 33.

To make the graph, I found three points, the vertex and both x-intercepts. I put them into Microsoft Excel, with the x coordinates in one column and the y coordinates in the other. I followed the steps necessary to create a scatterplot graph, and then added a trendline with a form of the standard equation.


Equations
The vertex form equation is:
y-57=-0.0523(x-0)^2

This equation was easy to do because I knew the vertex (0,57) from the beginning. However, I didn't know what 'a' (-0.05) was equivalent to at first.
So I did this:
0-57=a(33-0)^2
I filled in the x and y coordinates with one of the x-intercepts, (33,0). Then I simply solved for a, and ended up with this:
-57=1,089a
-0.0523=a
Of course, I rounded 'a' down quite a bit.

The standard form equation is:
y=-0.0523x^2+57
This was simple enough to figure out. I took the vertex equation and used it to solve for y.
-0.0523(x-0)^2=-0.0523x^2 - 0
y=-0.0523x^2 - 0+57
y=-0.0523x^2+57
This was by far the most easy of the equations.

The factored equation is:
-0.0523(x+33)(x-33)

Now would be a good time to talk about the discriminant. The discriminant of my parabola is not rational.
0^2 - 4(-0.0523)(57) = 11.9244
This makes it difficult to factor my standard equation, because there are no rational numbers that go into -0.0523 or 33. There is no rational factored equation for my parabola but I can make an estimate.


One way to show how this equation works is to multiply the factored equation back into standard form. When you multiply -0.0523 by (x+33) you get:
(-0.0523x - 1.7259)(x-33)
-33 * -0.0523x is positive. -1.7259x is negative. -33 * -0.0523x is 1.65x. Technically, 1.65x and -1.7259x cannot cancel each other out, but since I did not use all the decimal values of -0.0523, I have to believe they would be close enough to do so if had I had used them all.
When you simplify this equation and round down a few of the answers, you get:
-0.0523x^2+57

It's not exact, but it works well enough for this particular parabola.

Solution
For my solution, I decided to do the problem Mrs. Shutters suggested. Work needs to be done on the Eiffel Tower. The workmen's scaffolding is only 75 feet tall. Within my parabola, when does y equal 75 feet?
My parabola is 57 meters tall. Therefore, it would be easiest to convert 75 feet into meters.
75 feet=22.86 meters
Since the y axis is my axis of symmetry and my parabola faces down, I can use my standard form equation to figure this out.
First I replace y with 22.86:
22.86=-0.0523x^2+57
Now I just have to solve for x.
-0.0523x^2=22.86-57
-0.0523x^2=-34.14
x^2=-34.14/-0.0523
x=25.549 (approx.)

So, the x values with a y value of 22.86 meters are 25.549 and (because the y axis is the axis of symmetry) -25.549.

YAY!!