Gateshead Bridge

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An application of quadratic functions based on the Gateshead Bridge in the UK city of Newcastle. 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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1. Gap Width Number entry
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2. Gap Height Number entry
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The arch of the Gateshead Millenium Bridge in the United Kingdom (see illustration), forms a parabola which can be described by the equation

$y=-\frac{2}{125}x^2+\frac{210}{125}x+\frac{9}{10}$

where $y$ is the height of the arch above the water line in metres and $x$ is the horizontal distance from the left end of the arch in metres.

What is the width of the arch at it's base? (You may asume that the height of the base of the arch is the same on both sides of the river.) Show full working on your handwritten working.

Width of arch = Width metres

What is the height of the top of the arch above the water line? Show full working on your handwritten working.

Height of arch = Height metres

Advice

Width of river

To find the width of the river first note that at $x=0$ $y=\frac{9}{10}$ . At the opposite bank we will also have $y=\frac{9}{10}$ , hence we need to solve

$\frac{9}{10}=-\frac{2}{125}x^2+\frac{210}{125}x+\frac{9}{10}$

$0=-\frac{2}{125}x^2+\frac{210}{125}x=\frac{2}{125}x\left(105-x\right)$
This will occur at

$105-x=0$
that is

$x=105$ . Hence the width of the river is

$105$ metres.

Height of arch

We need to find the maximum value of the quadratic equation $y=-\frac{2}{125}x^2+\frac{210}{125}x+\frac{9}{10}$ .
The maximum height will occur at $\displaystyle{x=\frac{-b}{2a}}$ , that is $\displaystyle{x=\frac{210\times 125}{2\times 2\times 125}= 52.5}$ m.
The maximum height is

$\begin{align}h&=c-\frac{b^2}{4a}\\&=\frac{9}{10}+\frac{\left(210/125\right)^2}{8/125}\\&= 45 \quad\text{metres}\end{align}$
(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.

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Don Shearman 10 months, 4 weeks ago

Don Shearman 3 years, 7 months ago

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