Question

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.

Answer 1

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

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

Answer 2

The arch is above the road for a horizontal distance of 502.37 meters.

Explanation:

The Arch is modeled by

`y=-0.00211x^(2) +1.06x`

Which represents a parabola. Remember that quadratic equations represent parabolas.

Assuming that the road is the x-axis, we can get the answer by using
`y=0`, and solving the equation for
`x`

`0=-0.00211x^(2) +1.06x\\x(-0.00211x+1.06)=0`

Now, we use the zero property

`x_(1) =0\\-0.00211x_(2)+1.06=0 \implies x_(2)=(-1.06)/(-0.00211) \approx 502.37`

Therefore, the arch is above the road for a horizontal distance of 502.37 meters.