The Golden Gate Bridge is supported by two huge cables...

The Golden Gate Bridge is supported by two huge cables strung between the towers at each end of the bridge. The function $$f(x)=.0001193 x^{2}-.50106 x+526.113$$ gives the approximate height of the cables above the roadway at a point on the road $x$ feet from one of the towers. The cables touch the road halfway between the two towers. How far apart are the towers?

The Golden Gate Bridge is supported by two huge cables strung between the towers at each end of the bridge. The function $$f(x)=.0001193 x^{2}-.50106 x+526.113$$
gives the approximate height of the cables above the roadway at a point on the road $x$ feet from one of the towers. The cables touch the road halfway between the two towers. How far apart are the towers?

Labs

Want to see this concept in action?

NEW

Explore this concept interactively to see how it behaves as you change inputs.

View Labs

Key Concepts

Roots of Quadratic Equations

The roots of a quadratic equation are the x-values where the function equals zero, meaning the graph of the quadratic intersects the x-axis. Finding these roots is often accomplished through factoring, completing the square, or applying the quadratic formula. In physical applications, the roots can represent significant points such as the positions where a structure or motion meets a certain reference level.

Symmetry in Parabolas

Parabolas display an inherent symmetry about a vertical line known as the axis of symmetry, which runs through the vertex of the parabola. This symmetry implies that the two roots of the quadratic function, when they exist, are equidistant from the axis. In many practical problems, such as determining distances in bridge designs, recognizing this symmetry helps in setting up equations and finding unknowns, like the span between supports.

Quadratic Functions

Quadratic functions are polynomial functions of degree two that typically graph to form a curve known as a parabola. They are central in modeling real-world situations where relationships have a maximum or minimum value, such as the height of a cable or the trajectory of a projectile. Their standard form, ax² + bx + c, allows for analysis of key features like the vertex, axis of symmetry, and roots.

Transcript

Please give Ace some feedback