Chapter 7.3, Problem 57E

To determine

To find: the distance between the transmitters.

Answer to Problem 57E

183 miles

Explanation of Solution

Given information:

Let the angle of rays from ship to transmitters be $\theta$ , the distance of ship from one transmitter is $a$ and from another transmitter be $b$ .

  $\begin{array}{c}a=70\text{ miles}\\ b=130\text{ miles}\\ \theta =\text{130}°\end{array}$

Calculation:

Let the distance between two transmitters be $c$ . The value of $c$ an be evaluated by using the law of cosine as shown below.

  $\begin{array}{c}{c}^2=a^2+b^2-2ab\cos \theta \\ c=\sqrt{a^2+b^2-2ab\cos \theta }\left[\text{Taking square roots both sides}\right]\\ =\sqrt{{\left(70\right)}^2+{\left(130\right)}^2-2\left(70\right)\left(130\right)\cos \text{130}°}\left[\text{Put }a,b\text{ and }\theta \right]\\ =\sqrt{3349.87345}\\ =183\text{ miles}\end{array}$

Thus, the distance between two transmitters is 183 miles.