Question

The main cable of a suspension bridge forms a parabola, described by the equation y = a(x - h)2 + k, where y is the height in feet of the cable above the roadway, x is the horizontal distance in feet from the left bridge support, a is a constant, and (h, k) is the vertex of the parabola. At a horizontal distance of 30 ft, the cable is 15 ft above the roadway. The lowest point of the cable is 6ft above the roadway and is a horizontal distance of 90 ft from the left bridge support.Which quadratic equation models the situation correctly?
The main cable attaches to the left bridge support at a height of ft.
The main cable attaches to the right bridge support at the same height as it attaches to the left bridge support. What is the distance between the supports?

Answer 1

Refer to the diagram shown below.

When x = 30 ft, the cable is at 15 ft, therefore y(30) = 15.
That is,
a(30 - h)² + k = 15            (1)

Also, because the distance between the supports is 90 ft, therefore
y(0) = 6 ft, and y(90) = 6 ft
That is,
a(-h)² + k = 6                 (2)
a(90 - h)² + k = 6          (3)

From (2) and (3), obtain
a(90 - h)² = ah²
90² - 180h + h² = h²
180h = 90²
h = 45 ft.

From (1) and (2), obtain
225a + k = 15
2025a + k = 6

Therefore
1800a = -9
a = - 0.005
k = 15 - 225(-0.005) = 16.125 ft

Answer:
The equation for the cable is
y = - 0.005(x - 45)² + 16.125 

A graph of the solution verifies that the solution is correct.

Answer 2

Answer: The equation is $y=\frac{1}{400}(x-90)^2+6$. The main cable attaches to the left bridge support at a height of 26.25 ft. The distance between the supports is 180 ft.

Explanation:

The standard form of a parabola is,

$y=a(x-h)^2+k$

Where, (h,k) is vertex.

It is given that at a horizontal distance of 30 ft, the cable is 15 ft above the roadway.If means f(30)=15.

The lowest point of the cable is 6 ft above the roadway and is a horizontal distance of 90 ft from the left bridge support. It means the vertex is (90,6).

The equation can be written as,

$y=a(x-90)^2+6$

We have, f(30)=15.

$15=a(30-90)^2+6$

$9=a(-60)^2$

$a=\frac{1}{400}$

So, the equation of the parabola is,

$y=\frac{1}{400}(x-90)^2+6$

To find the height of main cable at left bridge support put x=0.

$y=\frac{1}{400}(0-90)^2+6$

$y=26.25$

So the height of main cable at left bridge support is 26.25 ft.

The parabola is symmetric along the axis of symmetry x=90. Since the distance of left bridge support from the axis of symmetry is 90, therefore, the distance of right bridge support from the axis of symmetry is also 90.

The total distance between both bridge support is,

$90+90=180$

Therefore, the distance between both bridge supports is 180 ft.