Arc d'Triomphe

Parabola Project

Rivka A.


The Arc de Triomphe is world's largest triumphal arch and it is located in Paris, France. The monument surmounts the hill of Chaillot at the center of a star-shaped configuration of 12 radiating avenue. Construction on the Arc de Triomphe began in 1806 to commemorate Napoleon Bonaparte's victories throughout Europe and it was designed by Jean François Thérèse Chalgri. The design of the arch is based on the Arch of Titus in Rome, Italy. The Arc de Triomphe stands at 51m (167 ft) high and 45m (148 ft) wide. The monument is so large that a daredevil pilot once flew his plane between the arches. The arch is an emblem of French patriotism, it honors the brave soldiers who fought for France during the French revolution. The structure has four huge relief sculptures, commemorating the emperor's battles.
archoftitus.jpg
This is the Arch of Titus. As you can see it's quite similar to the Arc de Triumphe, that's because the design of le Arc de Triumphe was based of of the Arch of Titus.

france-arc-de-triomphe.jpg
This is what the Arc de Triumphe looks like





The equations of the parabola are
y-95.8=-0.027(x-0)^2 Vertex form
y=-0.027x^2+95.8 Standard form
y=-0.027(x-59.63)(x+59.63) Factored form

I got the vertex form from knowing the length and width of the arc. After putting it in vertex form, i simplified it and got the standard form. With the standard form i was able to factor it creating the factored form. After i figured out the equations i inserted them into Grapher forming the parabola below.

Arc_de_triumphe_parabola.jpg
This is the parabola graphed
external image images.jpg
The vertex of the parabola = (0, 95.8) The vertex is where the parabola curves. In the Arc de Triumphe, it's the highest point of the arch.
The discriminant = 10.35 because b^2 -4ac, 0+(-4)*(-0.027)*95.8, equals 10.35. The discriminant helps determine if the equation is factorable or not. If it's a perfect square then it's factorable.
The y-intercept = 95.8 The y-intercept is where the parabola crosses the y-axis. In this parabolas case, the y-intercept is the same as the vertex.
The x-intercepts = (0, -59.63) (0, 59.63) The x-intercepts are where the parabola crosses the x-axis, or in the Arc de Triumphe where the arches hit the ground. A way to figure out the x-intercepts in a parabola is to use the quadratic formula.
-b+/- √b^2 -4ac
2a
A tourist launches a ball straight up into the air under the Arc de Triumphe. It reaches up to about 20 feet in the air. Where on the x-axis does the ball cross?

20= -0.027x^2 + 95.8
0= -0,027x^2 + 75.8

0+/- √8.2
-0.054
0+/- 2.9
-0.054
= -53.7 and 53.7

The ball crosses the x-axis at -53.7 and 53.7 feet.