Chapter 4.8, Problem 5E

Solution

To find: The length of the wire and the height of the antenna.

The length of the wire and the height of the antenna are approximately 287.94 ft and 283.56 ft respectively.

Given information:

Given, a guy wire connects the top of an antenna to a point on level ground 50 ft from the base of the antenna. The angle of elevation formed by the wire is 80º.

  

Calculation:

The triangle representing the data is:

  

Using definitions of trigonometric functions for the triangle ABC:

  $\begin{array}{l}cosC=\frac{\text{adjacent side}}{\text{hypotenuse}}\\ \cos 80°=\frac{50}{l}\\ l=\frac{50}{0.173}\\ l\approx 287.94\end{array}$

  $\begin{array}{l}\tan C=\frac{\text{opposite side}}{\text{adjacent side}}\\ \tan 80°=\frac{h}{50}\\ h=50\left(5.67\right)\\ h\approx 283.56\end{array}$

The length of the wire and the height of the antenna are approximately 287.94 ft and 283.56 ft respectively.

Conclusion:

The length of the wire and the height of the antenna are approximately 287.94 ft and 283.56 ft respectively.