Question

1. a guy - wire to a tower makes a 72° angle with the ground. at a point 30 ft farther from the tower than the wire, but on the same side as the wire, the angle of elevation to the top of the tower is 30°. find the length of the wire.

Explanation:

Step1: Let the length of the wire be \( l \) and the distance from the base of the tower to the point where the wire touches the ground be \( x \).

We know that \(\sin72^{\circ}=\frac{h}{l}\), so \(h = l\sin72^{\circ}\), and \(\cos72^{\circ}=\frac{x}{l}\), so \(x = l\cos72^{\circ}\).
Also, \(\tan30^{\circ}=\frac{h}{x + 30}\), so \(h=(x + 30)\tan30^{\circ}\).

Step2: Substitute \(h\) and \(x\) expressions.

Since \(l\sin72^{\circ}=(l\cos72^{\circ}+ 30)\tan30^{\circ}\).
Expand: \(l\sin72^{\circ}=l\cos72^{\circ}\tan30^{\circ}+30\tan30^{\circ}\).
Rearrange: \(l(\sin72^{\circ}-\cos72^{\circ}\tan30^{\circ})=30\tan30^{\circ}\).
We know that \(\sin72^{\circ}\approx0.9511\), \(\cos72^{\circ}\approx0.3090\), \(\tan30^{\circ}=\frac{\sqrt{3}}{3}\approx0.5774\).
\(\sin72^{\circ}-\cos72^{\circ}\tan30^{\circ}\approx0.9511-0.3090\times0.5774\approx0.9511 - 0.1783=0.7728\).
\(30\tan30^{\circ}=30\times\frac{\sqrt{3}}{3}=10\sqrt{3}\approx17.32\).
Then \(l=\frac{30\tan30^{\circ}}{\sin72^{\circ}-\cos72^{\circ}\tan30^{\circ}}\approx\frac{17.32}{0.7728}\approx22.4\)

Answer:

The length of the wire is approximately \(22.4\) ft.