Problem 51 - Algebra and Trigonometry

Question

In a suspension bridge the shape of the suspension cables is parabolic. The bridge shown in the figure has towers that are 600 m apart, and the lowest point of the suspension cables is 150 m below the top of the towers. Find the equation of the parabolic part of the cables, placing the origin of the coordinate system at the vertex. NOTE This equation is used to find the length of cable needed in the construction of the bridge.

Step-by-Step Solution

Answer

The equation of the parabola is \( y = \frac{1}{600}x^2 \).

Step 1: Identify Known Points

We are given that the suspension bridge forms a parabola with its vertex at the origin of a coordinate plane. The towers are 600 meters apart. This means the parabola passes through the points (300, 150) and (-300, 150), where the 150 meters is the height of the towers above the lowest point of the cable.

Step 2: Use Standard Form of Parabola

The equation of a parabola in standard form is given by \( y = ax^2 \). Since the vertex is at the origin (0,0), we will use this form to find the equation by substituting the known points of the parabola.

Step 3: Substitute Known Point into the Equation

Using one of the points (300, 150), substitute into the equation \( y = ax^2 \). This gives \( 150 = a(300)^2 \).

Step 4: Solve for the Constant 'a'

Solve the equation \( 150 = a(300)^2 \) to find the value of \( a \). This simplifies to \( 150 = 90000a \). Hence, \( a = \frac{150}{90000} = \frac{1}{600} \).

Step 5: Write the Complete Equation

Using the value of \( a \) found, the equation of the parabola is \( y = \frac{1}{600}x^2 \). This represents the shape of the suspension cables.

Key Concepts

Suspension Bridge

Suspension bridges are engineering marvels characterized by their distinctive architectural design, which includes cables hanging in a parabolic shape. These cables are not just for show; they play a crucial role in distributing weight and providing support to the bridge deck.

• The main cables in a suspension bridge are attached to tall towers, creating the familiar dip or sagging shape of the cables.
• The parabolic nature of these cables helps distribute the load effectively, minimizing stress on any single point.
• This efficient load distribution allows suspension bridges to span long distances without requiring multiple supports.

By placing the vertex of the parabolic cable at the lowest point between the towers, engineers can calculate the cable's exact length needed for construction. This provides the optimal balance between aesthetics and structural integrity.

Vertex of a Parabola

The vertex of a parabola is a significant point because it represents the highest or lowest point on the graph, depending on its orientation. In the context of a suspension bridge, the vertex is at the lowest point of the cable.

• For a parabola opening upwards, the vertex is the minimum point, ideal for modeling suspension bridges where cables dip down between towers.
• Positioning the coordinate system's origin at the vertex simplifies calculations, as the vertex coordinates become (0, 0).

In our suspension bridge example, determining the vertex position is essential for measuring how far the cable dips and ensuring the calculated lengths and angles are precise for effective bridge construction.

Equation of a Parabola

The equation of a parabola is a mathematical representation that describes its curvature. Specifically, if the vertex is at the origin, the equation simplifies to a form that is easily manipulated for construction purposes.

• The standard form of a parabola with a vertex at the origin is given by \( y = ax^2 \).
• The parameter \( a \) determines the "width" and direction of the parabola; a smaller \( a \) creates a wider parabola, which is fitting for our cable arrangement.

To find the parabola's equation that models the suspension bridge cables, plug in known points through which the curve passes. Solving the resulting equations allows engineers to have precise data on the cable length and curve, ensuring that every component of the bridge is crafted to specification. This equation isn't just a formula; it's a critical tool in the bridge's design and construction.