Problem 72 - Precalculus

Question

Each cable of the Golden Gate Bridge is suspended (in the shape of a parabola) between two towers that are 1280 meters apart. The top of each tower is 152 meters above the roadway. The cables touch the roadway midway between the towers. (a) Draw a sketch of the bridge. Locate the origin of a rectangular coordinate system at the center of the roadway. Label the coordinates of the known points. (b) Write an equation that models the cables. (c) Complete the table by finding the height \(y\) of the suspension cables over the roadway at a distance of \(x\) meters from the center of the bridge.

Step-by-Step Solution

Answer

The equation that represents the cables of the bridge is \(y = (152/640^2) * x^2\). The height of the cables at any given distance \(x\) from the center of the bridge can be calculated by substituting \(x\) into the equation.

Step 1: Sketching and Noting Co-ordinates

Let's sketch the bridge with a rectangular coordinate system located at the center of the roadway. The top of the towers are situated at \((640,152)\) and \((-640,152)\) and the center of the roadway where the cables touch is \((0,0)\). Take note of these coordinates.

Step 2: Forming Equation

The equation that describes a parabola is \(y = ax^2 + bx + c\). In this scenario, c = 0 since the parabola crosses the origin. As the parabola is symmetrical with respect to the y-axis, the term bx equals 0, thus eliminating it from the equation. Plugging the known points we get \(152 = a*640^2\), thus simplifying, we find \(a = 152/640^2\). So the equation of parabola is \(y = (152/640^2) * x^2\)

Step 3: Calculating Heights

To find the height of the suspension cables over the roadway at a distance x from the center of the bridge, substitute x into the equation derived from step 2. This forms the following instructions: \n\(y_{-300} = (152/640^2) * (-300)^2\)\n\(y_{-200} = (152/640^2) * (-200)^2\)\n\(y_{-100} = (152/640^2) * (-100)^2\)\n\(y_{100} = (152/640^2) * (100)^2\)\n\(y_{200} = (152/640^2) * (200)^2\)\n\(y_{300} = (152/640^2) * (300)^2\)

Key Concepts

Understanding the Coordinate System

A coordinate system is like an address system for points in space. It helps us locate precise points by using numbers

• The horizontal line is called the x-axis.
• The vertical line is the y-axis.
• Where these axes intersect is called the origin, denoted as (0,0).

In this exercise, the origin is placed right at the center of the Golden Gate Bridge's roadway.

This is a smart choice because it places equal parts of the bridge on either side. The coordinate (640, 152) represents the top of one tower, while (-640, 152) is the other. These coordinates give us the tower positions in relation to the origin, showing the height of the towers above the road and their distance from the center.

Writing the Equation of a Parabola

Parabolas have a unique U-shaped curve, represented by an equation in the form: \[y = ax^2 + bx + c\]When a parabola is symmetrical around the y-axis, as in the case for the cables of the Golden Gate Bridge, it simplifies to - \[y = ax^2 + c\] since \(b\) becomes zero. - Here, \(c = 0\) because the parabola passes through the origin. The golden rule for creating a precise equation is to use known points, like

• The origin at (0,0),
• The towers at (640,152) and (-640,152).

When we plug in any of these tower points into our equation, it lets us find \(a\): \[a = \frac{152}{640^2}\].This results in \[y = \frac{152}{640^2}x^2\], perfectly modeling the cables' curve.

The Golden Gate Bridge and Its Parabolic Shape

The Golden Gate Bridge, with its stunning suspension design, is a marvel of engineering.

Suspension cables form a parabola, a perfect choice due to its natural ability to evenly distribute weight.

• In this problem, the cables hang like gentle arcs between two towers, reaching their lowest point right at the bridge's center.
• This shape isn't just for aesthetic appeal; it provides strong structural support too!

By modeling this cable shape mathematically with a parabola, engineers ensure the bridge can stand the test of time. Knowing the properties of a parabola lets us calculate how high the cable is at any point along the roadway, assisting with both design and construction.

Thus, understanding parabolas isn't only important for math homework—it's crucial for creating monumental feats like the Golden Gate Bridge.