Question

a 192 ft tower is located on the side of a mountain that is inclined 27° to the horizontal. a guy wire is to be attached to the top of the tower and anchored at a point 58 ft downhill from the base of the tower. find the shortest length of wire needed.

note: the picture is not drawn to scale.

length of guy-wire = ft

enter your answer as a number; your answer should be accurate to 2 decimal places.

question help: video

Explanation:

Step1: Find the included angle

The mountain is inclined \(27^{\circ}\) to the horizontal. The tower is vertical. So the included angle \(\theta\) between the tower (\(a = 192\) ft) and the downhill distance (\(b=58\) ft) is \(180^{\circ}- 27^{\circ}=153^{\circ}\).

Step2: Apply the Law of Cosines

The Law of Cosines states that for a triangle with sides \(a\), \(b\), and \(c\) and included angle \(\theta\), \(c^{2}=a^{2}+b^{2}-2ab\cos\theta\).
Here, \(a = 192\), \(b = 58\), and \(\theta=153^{\circ}\), \(\cos(153^{\circ})=\cos(180 - 27)^{\circ}=-\cos(27^{\circ})\approx - 0.891\).
Substitute the values into the formula:

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Answer:

\(245.10\)