Chapter 8.2, Problem 42AYU

To determine

Find the height of the bridge.

Answer to Problem 42AYU

  $\boxed{1053.14698ft}$

Explanation of Solution

Given information:

The highest bridge in the world is the bridge over the Royal Gorge of the Arkansas River in Colorado. Sightings to the same point at water level directly under the bridge are taken from each side of the $880ft$ long bridge, as indicated in the figure. How high is the bridge?

  

Calculation:

Consider the figure, $\Delta ACB,\Delta ACD$ are right triangles, $\Delta ABD$ is an oblique angle.

  $\angle ABD=65.5°,\angle BDA=69.2°,BD=880ft$ and $AC=h$ .

  $\begin{array}{l}\angle ABD+\angle BDA+\angle DAB=180°\\ \angle DAB=180°-\left(\angle ABD+\angle BDA\right)\\ \angle DAB=180°-\left(65.5°+69.2°\right)\\ \angle DAB=45.3°\end{array}$

In triangle $\Delta ABD$ apply sine law,

  $\begin{array}{l}\frac{\sin A}{BD}=\frac{\sin B}{DA}=\frac{\sin D}{AB}\\ \frac{\sin 45.3°}{880}=\frac{\sin 65.5°}{DA}=\frac{\sin 69.2°}{AB}\\ \frac{\sin 45.3°}{880}=\frac{\sin 65.5°}{DA}\\ DA=880\times \frac{\sin \left(65.5°\right)}{\sin \left(45.3°\right)}\\ DA=880\times \frac{0.90996}{0.7107995}\\ DA=1126.57045ft\end{array}$

In a $\Delta ACD$ using trigonometric ratio of acute angle for right triangle,

  $\begin{array}{l}\sin \left(69.2°\right)=\frac{AC}{DA}\\ \sin \left(69.2°\right)=\frac{h}{1126.57045}\\ h=\left(1126.57045\right)\times \sin \left(69.2°\right)\\ h=\left(1126.57045\right)\times 0.93483\\ h=1053.14698ft\end{array}$

Hence, the height of the bridge is $\boxed{1053.14698ft}$