For my Parabola Project, I chose the bottom of the Eiffel Tower for my parabola.
external image eiffel-tower-bottom.jpg
The Eiffel Tower was built in 1889 by Gustave Eiffel, for the 1889 World Fair. It is in Paris, France. It is the tallest building in the city of Paris. Also, it is the most-visited paid monument in the world. It was the tallest building in the world from 1889 to 1930. It is an observation and radio broadcasting tower, and it is 1,063 feet including its spire.

Its maximum width is 410 feet, and its maximum length is 423 feet. In clear weather, you can see 43 miles from the top of the tower. Over 200,000,000 people have visited the Eiffel Tower. Its bottom is a parabola which opens down. The vertex of the parabola is also the y-intercept. The tower actually has multiple parabolas, but I chose this one.


The vertex of my parabola is (0, 189.07). I got this by finding out the height and width of the parabola. The height is 243.6 feet, and the width is 189.07 feet. I then put my parabola on a x and y-axis, which I put on the bottom of the parabola and down the middle of the parabola, respectively. I then found that the vertex is (0, 189.07) by putting that point on my parabola, which was now on the x-and y-axis. The vertex, in this case, signifies the middle of the top of the base of the Eiffel Tower. The vertex is a maximum, and the equation for the axis of symmetry is x=0. The x-intercepts are 121.8 and -121.8, because they are half of 243.6, and the y-intercept is 189.07. The equation for my vertex is y-189.07=a(x-0)^2, or y-189.07=ax^2.
The three quadratic equations are:
standard form: y=-0.0127x^2+189.07
vertex form: y-189.07=-0.0127x^2
factored form: y=-0.0127(x-121.8) (x+121.8)
I got these by figuring out the x-intercept, y-intercept, and vertex, and the 2 other points. The x-intercepts are 121.8 and -121.8, the y-intercept is 189.07, and the 3 points are (0, 189.07), (121.8, 0), and (-121.8, 0). I put these into Microsoft Excel, and found the trend line and graph, and the equation. Since the equation had a=0.0127 and b=0, that solved a and b for me. I put all of these into the standard form, vertex form, and factored form regular equations, and got these three equations.
chart.png
This is the graph of my equation. As you can see, the y-intercept is 189.07 and the x-intercepts are -121.8 and 121.8. You can also see that the vertex is (0, 189.07). The y-intercept describes when the arch reaches its top middle point.

For the discriminant, I used b^2-4ac, which would be, using the standard form of the equation, 0^2- (4) (-0.0127)(189.07)
The discriminant using this, is 0+9.604756, which is 9.604756. This is not a rational number. Therefore, the parabola cannot be factored over the integers. However, there are 2 solutions, although they are not rational.

There are 2 x-intercepts to the parabola. They are about 121.8 and about -121.8. They are not rational numbers. I know this because the discriminant is positive, but is not a rational number. I found them by doing b^2-4ac to find the discriminant, and figured out the actual numbers by graphing the parabola.They signify the points at which the parabola touch the ground. The factoring is related because the positivity/negativity and rationality/irrationality of the discriminant affects the amount of x-intercepts.

Suppose someone wanted to illegally build a stairway from 15 feet above the Eiffel Tower's base at night. To connect it to the ground, however, he needs to figure out where he could put the staircase to satisfy his obsession with having to use the parabola of the tower perfectly. In other words, when is the tower fifteen feet from the ground in relation to the base of the vertex of the tower?

To do this, I have to use the quadratic formula: -b +- sqrt(b^2-4ac) all divided by 2a. I know a, b, and c from the standard equation, with is -0.0127x^2+189.07. Therefore this means that 15=-0.0127x^2+189.07. I first subtract 15 from both sides, making it 0=-0.0127x^2+174.07, and now I can use the quadratic formula. Since b is zero, that means that -b is 0, so the square root of b^2-4ac is all that matters. b^2 is also zero, meaning that so far we have 0+-sqrt(0-4ac) 4 times 174.07 is 696.28, and 696.28 times -0.0127 is -8.842756. 0--8.842756 is 8.842756, so we need to find the square root of that. The square root of 8.842756 is 2.97367718, so we now have 0+-2.97367718/ divided by 2a. Our two things to divide by 2a are now 2.97367718 and -2.97367718. 2 times -0.0127 (2a) is -0.0254. So we now have 0+-2.97367718 divided by -0.0254. 2.97367718/-0.0254 is about -117, and -2.97367718/-0.0254 is about 117. Therefore, when the height of the parabola is 15, then the parabola is at 117 and -117.

Therefore, the two answers are (117,15) and (-117,15). This is related to the work we have done in chapter twelve in that we are figuring out parabolas based off of the quadratic formula, which is one of the things we did in Chapter 12.

For the focus and directrix of my parabola, I found that the focus is (x-value of vertex, K+P) I also found that P=1/4A, a being from the vertex form of the equation. Since I know that a=-0.0127, and 4 times that would be -0.0508. So 1/-0.0508 equals -19.685039. Since I also know that k=189.07. I can now find the focus. 189.07-19.685039=169.384961, and since the x-value of the vertex is zero, the focus is (0,169.384961).

For the directrix, I had to find (x-value of vertex, K-P). Since P is negative, and minus a negative is plus something, I now just do 189.07 plus 19.685039, which is 208.755039, so the directrix is (0,208.755039).