A telecommunications company connects two remote cell towers, Tower X and Tower Y, to a ce… — ACT

A telecommunications company connects two remote cell towers, Tower X and Tower Y, to a central relay station R. The cable path from Relay Station R to Tower X is $7$ kilometers long, and the cable path from Relay Station R to Tower Y is $8$ kilometers long. The angle formed between the two cables at Relay Station R, $\angle XRY$ , measures $120^\circ$ . A straight wireless backup link is established directly between Tower X and Tower Y. What is the distance, in kilometers, of this direct wireless link?

Answer: 13 km

Answer

The distance of the direct wireless link between Tower X and Tower Y is 13 kilometers.

Using the Law of Cosines with two side lengths of 7 km and 8 km and an included angle of 120° gives

XY^2 = 7^2 + 8^2 - 2(7)(8)\cos(120^\circ) = 49 + 64 - 112(-0.5) = 169

. Taking the square root gives 13 km.

Step-by-Step Solution

Identify known components of the triangle formed by the relay station and towers

Side

RX = 7\text{ km}

, side

RY = 8\text{ km}

, and included angle

\angle R = 120^\circ

Two sides and the included angle (SAS) are known, indicating that the Law of Cosines must be used to find the third side.

Apply the Law of Cosines formula for side

XY

XY^2 = 7^2 + 8^2 - 2(7)(8)\cos(120^\circ)

The Law of Cosines relates three sides of a triangle to the cosine of one of its angles:

c^2 = a^2 + b^2 - 2ab\cos(C)

Calculate the numeric value of

XY^2

XY^2 = 49 + 64 + 56 = 169

Since

\cos(120^\circ) = -0.5

, the term

-2(56)(-0.5)

evaluates to

+56

Take the principal square root of

169

XY = 13\text{ km}

Distance must be a positive length.

Key Concept

Law of Cosines (Side-Angle-Side configuration)

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