Gladesville Bridge

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An application of quadratic functions based on the Gladesville Bridge in Sydney, Australia. Student is given an equation representing the arch of the bridge and asked to find the height of the arch and the width of the river. Requires and understanding of the quadratic function and where and how to apply correct formulae.

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Don Shearman

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Don Shearman 10 months, 4 weeks ago

Published this.

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Don Shearman 3 years, 7 months ago

Created this.

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2. Gap Height Number entry
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The Gladesville Bridge is an arch bridge which spans the Parramatta River near Sydney. The bottom of the arch of the bridge (see illustration) can be modeled by the equation

$y=\frac{180}{305}x(1-\frac{1}{305}x)$ where

$y$ is the height of the arch above the water line in metres and

$x$ is the horizontal distance from the left bank of the river in metres.

What is the width of the river? (Hint: the arch is at the edge of the river on either side) Show full working in your handwritten working.

Width of river = Width metres

If the roadway is $3$ m above the the top of the arch, what is the maximum height of the road above the water line? Show full working in your handwritten working.

Height of roadway = Height metres

Advice

Width of river

To find the width of the river first note that at $x=0$ $y=0$ . At the opposite bank we will also have $y=0$ , hence we need to solve

$0=\frac{180}{305}x(1-\frac{1}{305}x)$
This will occur at

$1-\frac{1}{305}x=0$
that is

$x=305$ . Hence the width of the river is

$305$ metres.

Height of roadway

We need to find the maximum value of the quadratic equation $y=\frac{180}{305}x(1-\frac{1}{305}x)$ . Expanding this gives us the equation

$y=-\frac{180}{93025}x^2+\frac{180}{305}x$
The maximum height will occur at

$\displaystyle{t=\frac{-b}{2a}}$ , that is

$\displaystyle{t=\frac{180\times 93025}{305\times 2\times 180}= 152.5}$ m.
The maximum height is

$\begin{align}h&=c-\frac{b^2}{4a}\\&=0+\frac{\left(180/305\right)^2}{4\times180/93025}\\&= 45 \quad\text{metres}\end{align}$
(This value could also be found by substituting the value of

$x$ found above into the function.)
Hence the roadway will be

$45 + 3 =48$ metres above the water line.

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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.

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Don Shearman 10 months, 4 weeks ago

Don Shearman 3 years, 7 months ago

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