Problem 55 - Algebra and Trigonometry

Question

Suspension Bridge In a suspension bridge the shape of the suspension cables is parabolic. The bridge shown in the figure has towers that are 600 \(\mathrm{m}\) apart, and the lowest point of the suspension cables is 150 \(\mathrm{m}\) 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. INote: 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 parabolic cable is \( y = \frac{1}{600}x^2 \).

Step 1: Understanding the Parabolic Shape

Since the cables take a parabolic shape, we denote the parabola by the general equation \( y = ax^2 + bx + c \). Given that the vertex is at the origin \((0,0)\), the equation simplifies to \( y = ax^2 \). We know that the lowest point of the cable is at the vertex.

Step 2: Identifying Key Points

With the bridge between its towers being 600 meters apart, and knowing that the vertex is the lowest point, we find that key points on the parabola are \((-300, 150)\) and \((300, 150)\), where the cables connect to the top of the towers. These points are symmetrically placed around the origin.

Step 3: Substituting the Points Into the Parabolic Equation

Using the point \((300, 150)\), substitute into the simplified parabolic equation \( y = ax^2 \) to determine \( a \):\[ 150 = a(300)^2 \]Solving this gives us \[ a = \frac{150}{90000} = \frac{1}{600} \].

Step 4: Writing the Equation

Once we know \( a = \frac{1}{600} \), substitute back into the parabolic equation to find the specific equation for the cable's shape: \( y = \frac{1}{600}x^2 \).

Key Concepts

Suspension Bridge

Suspension bridges are engineering marvels that use parabolic cables to distribute their weight and tension across large distances. The cables in a suspension bridge hang in a parabolic curve, which efficiently transfers loads from the bridge deck to its towers, then down to the anchoring foundations. This curve resembles a mathematical parabola, an essential feature in designing and constructing these iconic structures. By understanding the parabolic shape of the cables, engineers can calculate vital parameters, like the length of the cable or the height of the towers, ensuring the bridge's structural integrity and safety.

Coordinate System

A coordinate system is a framework used to determine the position of points or objects in a given space. In the context of the suspension bridge problem, we set the origin at the vertex of the parabola. This choice simplifies the mathematical calculations, allowing the equation of the parabola to be easily managed. A coordinate plane consists of an x-axis (horizontal) and y-axis (vertical), intersecting at the origin point (0,0). When mapping the suspension bridge, the x-coordinate represents the distance from the center of the bridge to any point along its span, and the y-coordinate depicts the height above or below this central point.

Vertex (0,0)

The vertex of a parabola is its highest or lowest point, depending on its orientation. For a suspension bridge, the vertex is the lowest point of the suspension cable, located at the middle of the bridge between the towers. When positioned at (0,0) in a coordinate system, it provides a reference point that simplifies the equation of a parabola to the form of \( y = ax^2 \). As it is also the point of symmetry, it helps in designing structures that need equal distribution on either side. Understanding the role of the vertex in the parabolic equation is essential to both accurately formulating the equation and constructing the bridge.

Symmetry in Parabolas

Symmetry is a fundamental characteristic of parabolas, making them ideal for use in structures such as suspension bridges, where balance and even distribution of force are crucial. A parabola divides into two congruent halves, one on each side of its vertex, which is located at the origin in our bridge problem. This symmetry means that any property of the parabola on one side of the vertex is mirrored on the other side, creating a balanced structure. Recognizing this symmetry helps in calculating and understanding the parabolic path and tension of the cables, ensuring bridge stability and design efficiency.