Chapter 12.6, Problem 11P

To determine

Find the lengths of the cables.

Answer to Problem 11P

The lengths of the cables are about 31.6 m and 40.1 m.

Explanation of Solution

Given:

A vertical pole 20 m tall standing on a $15°$ slope is braced by two cables extending from the top of the pole to two points on the ground , 30 m up the slope and 30 m down the slope .

Calculation :

  

Find the measure of angle ADC , since the pole is at a right angle from the land,

  $\begin{array}{l}m\angle ADC+15°=90°\\ \Rightarrow m\angle ADC=75°\end{array}$

Now, the slope forms a line , so

  $\begin{array}{l}m\angle BDA+m\angle ADC=180°\\ \Rightarrow m\angle BDA=180°-75°\end{array}$

Using the Cosine Formula to find the third side of $\Delta ADC$ :

  $\begin{array}{l}AC=\sqrt{{20}^2+{30}^2-2\times 20\times 30\cos 75°}\\ \Rightarrow AC=\sqrt{1300-1200\cos 75°}=31.56\end{array}$

Using the Cosine Formula to find the third side of $\Delta ADB$ :

  $\begin{array}{l}AB=\sqrt{{20}^2+{30}^2-2\times 20\times 30\cos (180°-75°)}\\ \Rightarrow AB=\sqrt{1300+1200\cos 75°}\mathrm{........}[\cos (180°-x)=-\cos x]\\ \Rightarrow AB=40.132\end{array}$

So, the lengths of the cables are about 31.6 m and 40.1 m.