Question

You live near a bridge that goes over a river. The underside of the bridge is an arch that can be modeled with the function y = –0.000475x2 + 0.851x, where x and y are in feet. How high above the river is the bridge (the top of the arch)? How long is the section of bridge above the arch? (1 point)

The bridge is about 1,791.58 ft. above the river, and the length of the bridge above the arch is about 381.16 ft.

The bridge is about 1,791.58 ft. above the river, and the length of the bridge above the arch is about 895.79 ft.

The bridge is about 381.16 ft. above the river, and the length of the bridge above the arch is about 895.79 ft.

The bridge is about 381.16 ft. above the river, and the length of the bridge above the arch is about 1,791.58 ft.

Answer 1

The correct option is D.

Explanation:

The underside of the bridge is an arch that can be modeled with the function

`y=-0.000475x^2+0.851x`

Where, x is the horizontal distance from one end of the bridge (in feet) and y is height of the bridge above the river (in feet).

The leading coefficient is negative so it is a downward parabola.

If a function is
`y=ax^2+bx+c`, then its vertex is

`((-b)/(2a),f((-b)/(2a)))`

In the given function a=-0.000475 and b=0.851.

`(-b)/(2a)=(-0.851)/(2(-0.000475))=895.789`

`f((-b)/(2a))=f(895.789)=-0.000475(895.789)^2+0.851(895.789)=381.158\approx 381.16`

The vertex of the downward parabola is the point of maxima. The vertex of the function is (895.79,381.16).

Therefore the bridge is about 381.16 ft. above the river.

Substitute y=0 in the given function, to find the x-intercepts of the function.

`0=-0.000475x^2+0.851x`

`0=x(-0.000475x+0.851)`

Using zero product product property,

`x=0`

`-0.000475x+0.851=0`

`-0.000475x=-0.851`

`x=(-0.851)/(-0.000475)\approx 1791.58`

The x-intercepts of the function are 0 and 1791.58.

The time in which the section of bridge above the arch is the difference of x-intercepts because it is an downward parabola.

`1791.58-0=1791.58`

The time in which the section of bridge above the arch is 1791.58 ft.

The bridge is about 381.16 ft. above the river, and the length of the bridge above the arch is about 1,791.58 ft.

Therefore the correct option is D.

Answer 2

The height of bridge will be found as follows:
y=-0.000475x^2+0.851x
at maximum height, dy/dx=0
thus
dy/dx=-0.00095x+0.851=0
thus x=895.7895 =895.79 ft
at this point the maximum height will be:
h(x)=895.79 ft

The length of the section will be given by:

y=-0.000475x^2+0.851x
solving for the x-intercept we get:
-0.000475x^2+0.851x
thus x=0 or x=1791.58 ft
thus the correct answer is D