Question

a communications tower is located at the top of a steep hill, as shown. the angle of inclination of the hill is 70°. a guy wire is to be attached to the top of the tower and to the ground, 104 m downhill from the base of the tower. the angle formed by the guy wire is 14°. find the length of the cable required for the guy wire. note: the picture is not drawn to scale. length of guy-wire = m enter your answer as a number; your answer should be accurate to 2 decimal places. question help: worked example 1

Explanation:

Step1: Find the third angle of the triangle

In a triangle, the sum of angles is \(180^{\circ}\). Let the third angle be \(A\).
We know two angles: one is \(70^{\circ}\) and the other is \(14^{\circ}\). So \(A = 180^{\circ}-(70^{\circ}+ 14^{\circ})=96^{\circ}\)

Step2: Apply the Law of Sines

The Law of Sines states that \(\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}\). Let the length of the guy - wire be \(x\). We know that \(b = 104\) m, \(B=14^{\circ}\), and \(A = 96^{\circ}\)
By the Law of Sines \(\frac{x}{\sin70^{\circ}}=\frac{104}{\sin14^{\circ}}\)

$$x=\frac{104\times\sin70^{\circ}}{\sin14^{\circ}}$$

We know that \(\sin70^{\circ}\approx0.9397\) and \(\sin14^{\circ}\approx0.2419\)

$$x=\frac{104\times0.9397}{0.2419}$$

$$x=\frac{97.7288}{0.2419}\approx404.01$$

Answer:

\(404.01\)