Chapter 3.4, Problem 13AYU

To determine

Find the height of the cables at a point $100$ meters from the center.

Answer to Problem 13AYU

Theheight of the cables at a point $100$ meters from the center is $18.75ft.$

Explanation of Solution

Given:

Length of twin towers: $75$ meters and $400$ meters apart from road.

Calculation:

A suspension bridge with weight uniformly distributed along its length has twin towers that extend $75$ meters above the road surface and are $400$ meters apart.

The cables are parabolic in shape and are suspended from the tops of the towers. The cables touch the road surface at the center of the bridge.

Since there is no vertical shift and horizontal shift right $200$ part of the equation in vertex form will be as, $y=z{\left(x-200\right)}^2$ , where $z$ is the unknown amount of vertical compression.

Now to find the value of $z$ use an $x$ value where its corresponding $y$ value and plug it into the equation, such as the $x$ value $0$ with the corresponding $y$ value of $75.75=z{\left(0-200\right)}^2$ . Then solve for $z$ .

Now, $z=\frac{3}{1600}{\left(x-200\right)}^2$ .

Then simply substitute $100$ or $300$ for $x$ and it should equate to $18.75ft.$

Conclusion:

Hence, the height is $18.75ft.$