Hello, cocoknny!

An attendant in a lighthouse receives a request for aid

from a stalled aircraft located 15 miles due east of the lighthouse.

The attendant contacts a second boat located 14 miles from the lighthouse

on a bearing of N23°W.

What is the distance and the bearing of the rescue boat from the stalled craft? Code:

B *
* *
* *
* : *
*23°: *
14 * : * :
* : * :
*: * θ :
* - - - - - - - - - - - - - - - - - - - - *
L 15 A

The lighthouse is at $\displaystyle L$. .The stalled aircraft is at $\displaystyle A.;\;LA = 15$

The rescue boat is at $\displaystyle B.\;\;BL = 14,\;\angle BLA = 113^o$

We want the distance $\displaystyle AB$ and the bearing $\displaystyle \angle\theta.$

Law of Cosines: .$\displaystyle AB^2

;=\;14^2 + 15^2 - 2(14)(15)\cos113^o \;=\;585.107074$

. . Hence: .$\displaystyle \boxed{AB \:\approx\:24.2\text{ miles}}$

Law of Sines: .$\displaystyle \frac{\sin A}{14} \:=\:\frac{\sin113^o}{24.2}\quad\Rightarrow\quad\s in A \:=\:0.532523469$

Hence: .$\displaystyle \angle A \:\approx\:32.2^o\quad\Rightarrow\quad\boxed{\angl e\theta \:=\:57.8^o}$

Therefore, $\displaystyle B$ is $\displaystyle 24.2$ miles from $\displaystyle A$ at a bearing of $\displaystyle N57.8^oW$