Chapter 8.2, Problem 42AYU

Finding the Height of the Bridge over the Royal Gorge 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 880-foot-long bridge, as indicated in the figure. How high is the bridge?

Source: Guinness Book of World Records

To determine

To find: The height of the bridge over the Royal Gorge.

Answer to Problem 42AYU

Solution:

1053 feet

Explanation of Solution

Given:

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 880–foot–long bridge, as indicated in the figure. How high is the bridge?

Formula used:

$\text{tan} = opp/hyp$

$\angle \text{A} + \angle \text{B} + \angle \text{C} = {180}^{\circ }$

Calculation:

So from the triangle $\Delta \text{ACD}$

Lets say $\text{DC}$ distance is $\text{x}$ feet so the value of $\text{BD}$ will be ($880–\text{x}$) feet(we know that the distance from $\text{B}$ to $\text{C}$ is 880 feet totally

$\text{tan}69.2° = \frac{\text{AD}}{\text{DC}}$

$\text{AD} = \text{DC} \times \text{tan}69.2°$   –––––equation 1

From the triangle $\Delta \text{ABD}$

$\text{tan}65.5° = \frac{\text{AD}}{\text{BD}}$

$\text{AD} = \text{BD} \times \text{tan}65.5°$   –––––equation 2

From equation 1 and 2

$\text{AD} = \text{AD}$

So

$\begin{array}{l}\text{BD} \times \text{tan}65.5° = \text{DC} \times \text{tan}69.2°\\ \frac{\text{BD}}{\text{DC}} = \frac{\text{tan}69.2°}{\text{tan}65.5°} \approx 1.2\\ \frac{880 - \text{x}}{\text{x}} = 1.2\\ \frac{880}{\text{x}} - 1 = 1.2\\ \frac{880}{\text{x}} = 2.2\\ \text{X} = \frac{880}{2.2} = 400 \text{feet}\end{array}$

So now

$\begin{array}{l}\text{tan}69.2° = \frac{\text{AD}}{\text{DC}}\\ \text{AD} = \text{DC} \times \text{tan}69.2°\\ \text{h} = \text{x} \times \text{tan}69.2°\\ \text{h} = 400 \times 2.63\\ \text{h} = 1053 \text{feet}\end{array}$

so the height of the bridge is 1053 feet