Question

Two guys wires are attached to the top of a radio antenna 25m in height. The wires make angles of π/6 and π/4 with the ground, as shown. Determine the exact expression for the distance between the two anchor points of the wires, A and B.

Answer 1

To determine the distance of two anchor points A and B, first, we need to calculate the distance from each point to the base of the radio antenna, then, subtract the results.

To solve the distance of each anchor point to the antenna, we can use the tangent function.

`\tan \theta=\frac{\text{opposite side of the angle}}{\text{adjacent side of the angle}}`

Based on the given angles, its opposite side is the radio antenna with a height of 25 meters while its adjacent side is the distance from the anchor point to the base of the antenna which we want to solve.

Let's start with Point A. Plug in the given angle and the length of the opposite side to the function above.

`\tan (\pi)/(6)=(25m)/(x)`

Cross multiply and solve for x. (Make sure the calculator being used is set into Radians since the given angle is in radian.)

`\begin{gathered} x\tan (\pi)/(6)=25m \\ x=(25m)/(\tan (\pi)/(6))\Rightarrow x=\frac{25m}{\frac{\sqrt[]{3}}{3}}\Rightarrow x=25\sqrt[]{3}meters \end{gathered}`

Hence, the distance from Point A to the base of the radio antenna is 25√3 meters.

Now, let's move on to Point B. Use the tangent function still.

`\begin{gathered} \tan \theta=\frac{opposite\text{ side}}{\text{adjacent side}} \\ \tan (\pi)/(4)=(25m)/(x) \\ x=(25m)/(\tan (\pi)/(4))\Rightarrow x=(25m)/(1)\Rightarrow x=25meters \end{gathered}`

Hence, the distance from Point B to the base of the radio antenna is 25 meters.

To illustrate that, we have:

So, to get the distance between A and B, we simply subtract the distance.

`\begin{gathered} 25\sqrt[]{3}m-25m \\ or \\ -25+25\sqrt[]{3}meters \end{gathered}`

The distance between two anchor points A and B in exact expression is -25 + 25√3 meters.