Chapter 4.3, Problem 71E

a.

To determine

How long is the guy wire?

Answer to Problem 71E

  $\boxed{219.94ft.}$

Explanation of Solution

Given information:

A guy wire runs from the ground to a cell tower. The wire is attached to the cell tower $\text{15}0$ feet above the ground. The angle formed between the wire and the ground is ${43}^{\circ }$ (see figure).

How long is the guy wire?

Calculation:

It is given that a guy wire runs from the ground to a cell tower. The wire is attached to the cell tower $\text{15}0$ feet above the ground. The angle formed between the wire and the ground is $\theta ={43}^{\circ }$ .

We have to find the length of the wire.

Let $x$ be the length of the wire.

Now, $x$ is the length of the hypotenuse of the above triangle.

Length of the side opposite to $\theta ={43}^{\circ }$ is $\text{15}0$ feet.

Now, we know that,

  $\sin \theta =\begin{array}{c}\text{Opposite side to }\theta \\ \text{hypotenuse}\end{array}$

Substituting all the above values in this formula, we have,

  $\begin{array}{l}\sin {43}^{\circ }=\begin{array}{c}150\\ x\end{array}\\ x=150\times \sin {43}^{\circ }\\ x=150\times 0.0682\\ x=219.94\end{array}$

Hence, the length of the wire is $\boxed{219.94ft.}$ .

b.

To determine

Distance from the base of the tower is the guy wire anchored to the ground.

Answer to Problem 71E

  $\boxed{160.86ft.}$

Explanation of Solution

Given information:

A guy wire runs from the ground to a cell tower. The wire is attached to the cell tower $\text{15}0$ feet above the ground. The angle formed between the wire and the ground is ${43}^{\circ }$ (see figure).

How far from the base of the tower is the guy wire anchored to the ground?

Calculation:

It is given that a guy wire runs from the ground to a cell tower. The wire is attached to the cell tower $\text{15}0$ feet above the ground. The angle formed between the wire and the ground is $\theta ={43}^{\circ }$ .

We have to find the distance from the base of the tower is the guy wire anchored to the ground.

Let $y$ be the length of the wire.

Now, $y$ is the length of side adjacent to angle $\theta ={43}^{\circ }$ .

Now, we know that,

  $\tan \theta =\begin{array}{c}\text{Opposite side to }\theta \\ \text{adjacent side to }\theta \end{array}$

Substituting all the above values in this formula, we have,

  $\begin{array}{l}\tan {43}^{\circ }=\begin{array}{c}150\\ y\end{array}\\ y=\begin{array}{c}150\\ 0.933\end{array}\\ y=160.86\end{array}$

Hence, the distance from the base of the tower is the guy wire anchored to the ground is $\boxed{160.86ft.}$ .