Chapter 4.8, Problem 15E

Solution

To find: The height of the observation deck.

The height of the observation deck is approximately 52.35 ft.

Given information:

Given, the angle of elevation from an observer to the bottom edge of the Delaware River drawbridge observation deck located 200 ft from the observer is 30°. The angle of elevation from the observer to the top of the observation deck is 40°.

  

Calculation:

The triangle representing the data is:

  

Using definitions of trigonometric functions for the triangle ABC:

  $\begin{array}{l}\tan \left(\angle ABC\right)=\frac{\text{opposite side}}{\text{adjacent side}}\\ \tan \left(30°\right)=\frac{x}{200}\\ x=\frac{200}{\sqrt{3}}\end{array}$

Also for the triangle ABD,

  $\begin{array}{l}\tan \left(ABD\right)=\frac{\text{opposite side}}{\text{adjacent side}}\\ \tan \left(40°\right)=\frac{x+h}{200}\\ x+h=\left(200\right)\tan \left(40°\right)\end{array}$

Plugging the value of x :

  $\begin{array}{l}\frac{200}{\sqrt{3}}+h=\left(200\right)\tan \left(40°\right)\\ h=\left(200\right)\left(0.839\right)-\left(200\right)\left(0.577\right)\\ h\approx 52.35\end{array}$

The height of the observation deck is approximately 52.35 ft.

Conclusion:

The height of the observation deck is approximately 52.35 ft.