Golden Gate Bridge

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Don Shearman

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Terry Young

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Terry Young 6 months, 4 weeks ago

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The Golden Gate bridge in San Francisco is a large suspension bridge. The cable joining the two towers of the bridge (see illustration) can be modeled by the equation

$y=\frac{3}{8000}x^2-\frac{9}{20}x+155$ where

$y$ is the height of the cable above the bridge deck in metres and

$x$ is the horizontal distance from the left tower in metres.

What is the distance between the towers? (Hint: both towers are the same height. The left hand tower is at $x=0$ ) Show full working on your handwritten working.

Width between towers = width metres.

What is the minimum height of the cable above the deck of the bridge? Show full working on your handwritten notes.

Minimum height of cable above deck = Height metres.

Advice

To find the distance between the towers first note that at $x=0$ (the location of the first tower) the height of the cable is $\frac{3}{8000}\times 0^2-\frac{9}{20}\times 0+155=155$ m. To find the location of the other tower we need to find the other $x$ value with $y=155$ , that is we need to solve

$155=\frac{3}{8000}x^2-\frac{9}{20}x+155$
or equivalently

$\frac{3}{8000}x^2-\frac{9}{20}x=0.$
This can be done by factorising the above equation to

$\frac{3}{8000}x\left(x-1200\right)=0$
and reading the roots as

$x=0$ and

$x=1200$ . Hence the second tower is at a distance of

$1200$ m from the first tower.

We need to find the minimum value of the quadratic equation $y=\frac{3}{8000}x^2-\frac{9}{20}x+155$ . (The value will be a minimum since $a$ , the coefficient of $x^2$ is greater than $0$ )
The minimum height will occur at $\displaystyle{t=\frac{-b}{2a}}$ , that is $\displaystyle{t=\frac{9\times 8000}{20\times 2\times 3}= 600}$ m.
The minimum height is

$\begin{align}h&=c-\frac{b^2}{4a}\\&=155-\frac{\left(9/20\right)^2}{12/8000}\\&= 20 \quad\text{metres}\end{align}$
(This value could also be found by substituting the value of

$x$ found above into the function.)

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This question is used in the following exams:

SCC Class Test 2 by Hanh Vo in SCC PrelimMaths.
SCC Class Test 2 by Hanh Vo in SCC PrelimMaths.
Don's copy of Class Test 2 by Don Shearman in MfEP Progress Quizzes.
NCM4 TEST Regression Modeling Basics by Terry Young in Terry's workspace.

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