Golden Gate Bridge

Parabola Project

Abie F.


The parabola I chose is the Golden Gate Bridge.

The Golden Gate Bridge is an iconic American landmark. It is located in San Francisco, California. It stretches over the San Francisco bay's opening into the Pacific ocean. The Golden Gate Bridge is the second longest suspension bridge in the United States. 80,000 miles of wire help make up the structure, along with 1,200,000 rivets. The Golden Gate Bridge is also a great example of a parabola because of its slightly rounded shape. The Golden Gate bridge is not only a symbol of California, it is also a symbol of the United States.

golden_gate.png
A view of the parabola formed by the suspension cables.

The architect of the Golden Gate Bridge was Joseph B. Strauss.

The main suspension cables between the towers of the Golden Gate Bridge form a parabola that can be modeled by the quadratic function:

Y = 0.000112x2 + 220


where x is the horizontal distance from the middle of the bridge to the towers and y is the vertical distance from the water level to the bridge deck. The cables are connected to the towers at points that are 5000 feet above the road, and the road is about 220 feet above the mean water level.

A few Golden Gate Bridge facts to illustrate its size:

· Total length: Including approaches, 1.7 miles (8,981 feet or 2,737 m)
· Middle span: 4,200 feet (1,966 m).
· Width: 90 feet (27 m)
· Clearance above the high water (average): 220 feet (67 m)
· Total weight when built: 894,500 tons (811,500,000 kg)
· Total weight today: 887,000 tons (804,700,000 kg). Weight reduced because of new decking material
· Towers:
· 746 feet (227 m) above the water
· 500 feet (152 m) above the roadway
· Each leg is 33 x 54 feet (10 x 16 m)
· Towers weigh 44,000 tons each (40,200,000 kg).
· There are about 600,000 rivets in EACH tower.

As shown above in question 2, the span between the towers is 4,200 feet. Therefore, as shown in the table immediately above, the vertex or midpoint of the bridge span between the towers is at 2,100 feet. Using the parabolic equation for the Golden Gate Bridge, y=0.000112x2 + 220, we were able to calculate the span at the towers as being 713.92 feet above the water. We also know from the y-intercept that the bridge is 220 above water level.

golden_gate_graph.png

4. Give Equations For Your Parabola.
Standard Form: y=0.000112x2 + 0x + 220

Vertex Form: NEED HELP
Factored Form:

5. Give Details About Your Parabola.
a. Vertex: The vertex of a parabola is the point of intersection of a parabola with its axis of symmetry. I realized the vertex was when I was looking at the graph, when the axis of symmetry and the parabola met is where the vertex is. To find the vertex without looking at a graph you could make a table to determine the point of the axis of symmetry. The equation is a maximum because the coefficient of the squared term (.000112x2 ) is positive. The vertex signifies the mid-point of the bridge. The equation for the axis of symmetry is y = (x-h)2. Where the vertex is where x = h.

b. Y-Intercept: The Y-Intercept is the y-coordinate of a point where a graph intersects the Y-axis. I looked at the graph and I looked for the lowest point on the Y-axis and was 220. I would solve the equation for x=0. For the Golden Gate Bridge it signifies the height of the bridge over the water underneath.

c. Discriminant: The Discriminant is the value of b2 - 4ac in the quadratic equation ax2 + bx+ c = 0. In the equation for the Golden Gate Bridge parabola b=0, a=0.000112, and c=220. Because b^2-4ac is less than zero the equation has no real solutions, therefore it is not factorable over the integers.

d. X-Intercept: There is no x-intercept because the y-intercept is positive. You factor the equation and solve for x. Because the Golden Gate Bridge parabola has no real solutions, you cannot solve for x and that tells us that there are no x-intercepts, if there were x-intercepts in this case, this would have signified that the bridge had fallen down.

e. Solutions: ????



f. Focus & Directrix:

focus_dir.png
Parabolic curve showing directrix (L) and focus (F). The distance from a given point Pn to the focus is always the same as the distance from Pn to a point Qn directly below, on the directrix.

g. Conic Sections:

A parabola is the intersection of a cone with a plane.
conic1.png
Also, parabolas are conic sections.
conic2.png