Chapter 10.2, Problem 67AYU

Suspension Bridge The cables of a suspension bridge are in the shape of a parabola, as shown in the figure. The towers supporting the cable are 600 feet apart and 80 feet high.

If the cables touch the road surface midway between the towers, what is the height of the cable from the road at a point 150 feet from the center of the bridge?

To determine

To find: The height of the cable from the road at a point 150 feet from the centre of the bridge if the towers supporting the cable is 600 feet apart and 80 feet high and the cable touch the road midway between the towers.

Answer to Problem 67AYU

The height of the cable from the road at the point 150 feet from the center of the bridge is 20 feet.

Explanation of Solution

Given:

The cables of a suspension bridge are in the shape of a parabola, as shown in the figure. The towers supporting the cable are 600 feet apart and 80 feet high. The cable touches the road surface midway between the towers.

Formula used:

Vertex	Focus	Directrix	Equation	Description
$\left(0,0\right)$	$\left(0,a\right)$	$y=-a$	$x²=4ay$	Parabola, axis of symmetry is $y\text{-axis}$, opens up.

Calculation:

Let the vertex of the parabola be $\left(0,0\right)$ and the bridge (parabola) opens up.

The equation of the parabola is given by, $x²=4ay$.

From the given data, length of the bridge between the towers is 600 ft, Hence $\left(300,80\right)$ and $\left(-300,80\right)$ lies on the parabola.

Substituting the points in the equation $x²=4ay$, we get,

$300²=4*a*80$

$a=\frac{{300}^2}{4*80}$

Hence the equation of the parabola is,

$x²=4*\frac{{300}^2}{4*80}y$

$x²=1125y$

Parabolic representation is given below.

We have to find the height of the cable when it is at a distance of 150 feet from the centre of the bridge. Hence the point $\left(150, h\right)$ is on the parabola.

Substituting the point $\left(150, h\right)$ in the equation of the parabola, we get,

$x²=1125*h$

$150²=1125*h$

$h=150*\frac{150}{1125}$

$h=20$

The height of the cable from the road at the point 150 feet from the center of the bridge is 20 feet.