Sydney+Opera+House

I chose the Sydney Opera House, quite by chance. I am glad that i have it, however. It is a fairly easy structure to conceptualize a graph for especially since it was easy to find the Opera House's measurements.

The Sydney Opera House is located in Sydney, Australia. It is a beautiful building located right near the Sydney Harbor, right on the ocean. The Opera House was created to give Sydney it’s very own music venue, since prior to the Opera House construction, musical events were conducted in the Town Hall. The Opera House was opened to Queen Elizabeth on the 20th of October 1973. The Opera House cost so much to build that it was not fully paid for until 1975, two years after its opening. The Sydney Opera House was designed by a Danish architect named Jorn Utzon. Utzon took his inspiration from many places. He took some of his design inspiration from beech trees, and the waves, and the seating arrangement inside was taken from the Grecian theater. Utzon was not the only one to design the Opera House. Utzon also had the help of Ove Arup and Partners to solve the difficult problem of the roof. There were many geometric shapes that were taken into consideration, and the spherical one suggested by Utzon is the shape that was ultimately taken on.

If the Sydney Opera House was put onto a graph with the y-axis going straight down the middle, and the x-axis being the ground, then the Opera House would have a maximum y-intercept value of 185, since the height of the Opera House is 185m, the vertex is also at (0,185). The Opera House has two x-intercepts at (60,0) and (-60,0), because the y-intercept splits the graph in half, and the length from one side to the other is 120m long. The x-intercepts signify where the Opera House hits the ground, and the vertex signifies the highest point on the Opera House.


 * Equations for the Parabola Made by the Sydney Opera House**

How I found it- Once I found the vertex i plugged it into the equation. To solve for 'a' I found one of the x-intercepts and plugged //that// into the equation. After that it was a simple matter of isolating 'a' and the end result was a=-.0514.
 * VERTEX FORM--- y-185=-.0514(x-0)^2**

How I found it- Since I knew the vertex form, the y-intercept, and the point (60,0), it was easy to solve for 'b', which turned out to be b=0. Then I just substituted 'c' for the y-intercept, 185, and i substituted 'a' for the 'a' value in the vertex form.
 * STANDARD FORM--- y=-.0514x^2+185**

Factored form was the easiest, since I just plugged in my x-intercepts, and used the 'a' value from my previous two equations.
 * FACTORED FORM--- y=-.0514(x-60)(x+60)**

If the Quadratic Equation is used on the Sydney Opera House parabola when y=100 then the solutions of the parabola are -40.67 and 40.67. This is helpful, because when the Quadratic Equation is used on any of the y-values in the graph, then you can also see what the x-values are at that given y-value point. It helps to find the horizontal value when you know a vertical value.

The Discriminant of the Sydney Opera House Parabola is 38.036. The discriminant tells whether the equation is factorable over the integers, which my parabola is not. The discriminant also indicates how many x-intercepts there are, since my discriminant is positive, there are two x-intercepts.


 * EXTRA CREDIT---CONIC SECTIONS**

When a parabola is spun on an axis, it forms a cone. Conic sections are used to determine what a parabola's cone would look like without using a fancy computer program.


 * MATH PROBLEM**

The Sydney Opera House is going to be decorated for the coming of famous operatic singer Pari Bolai. They want to hang a gigantic banner in front of the opera house at 100 meters. Assuming that the banner will run straight across, and that it will be attached to the Opera House at two points, using the quadratic equation, find what two points the banner will be attached to when y=100. //The banner will be attached at points -40.67 and 40.67.//