Sydney harbor bridge height

Asked by maham237 @ 10/01/2021 in Mathematics viewed by 413 People

The arch of the Sydney Harbor Bridge in Sydney, Australia, can be modeled by y=−0.00211x2+1.06x, where x is the distance (in meters) from the left pylons and y is the height (in meters) of the arch above the water. For what distances x is the arch above the road? Round your answers to the nearest whole number.

Answered by maham237 @ 10/01/2021

Answer:

The arch above the road is 251 meters from the left pylons

Step-by-step explanation:

we have

$y=-0.00211x^{2}+1.06x$

This is a vertical parabola open downward (because the leading coefficient is negative)

The vertex is a maximum

The y-coordinate of the vertex represent the maximum height of the arch above the road

Remember that the x-coordinate of the vertex is the midpoint between the roots of the quadratic equation

One root is the origin (0,0)

Determine the second root

For y=0

$-0.00211x^{2}+1.06x=0$

$0.00211x^{2}=1.06x$

Simplify

$0.00211x=1.06$

$x=502$

Find the midpoint between the roots

$x_m=(0+502)/2=251$

so

The x-coordinate of the vertex is 251

therefore

The arch above the road is 251 meters from the left pylons

see the attached figure to better understand the problem

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