Answer:
198 miles
Explanation:
The distance from the starting point can be found by adding the two distance vectors. It can also be found by solving the triangle described by the points on the route.
Law of Cosines
The law of cosines can be used to solve a triangle when two sides and the included angle are known. It tells you ...
c² = a² +b² -2ab·cos(C) . . . . where C is the angle between sides 'a' and 'b'
In the geometry modeling the sailing route, the interior angle at the turn is ...
180° -(191° -78°) = 67°
Then the equation for the distance from start is ...
c² = 127² +209² -2·127·209·cos(67°) ≈ 39,067.65
c ≈ √39067.65 ≈ 197.66
The distance of the boat from its starting point is about 198 miles.
Vector sum
The sum of the two vectors can be found using a suitable calculator. (See the second attachment.) Or it can be calculated "by hand" by resolving the vectors to their components.
OA = 127(sin(78°), cos(78°)) = (124.22, 26.40)
AB = 209(sin(191°), cos(191°)) = (-39.88, -205.16)
OB = OA + AB = (124.22 -39.88, 26.40 -205.16) = (84.34, -178.76)
The magnitude of vector OB is ...
|OB| = √(84.34² +(-178.76)²) = √39067.65 ≈ 197.66
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Additional comment
Rounded values are shown above. Calculations were done with enough significant digits to ensure the results shown are accurate.