SOLUTION: Two Coast Guard stations, A and B, are on an east-west line and are 72 km apart. The bearing of a ship from station A is N 54o E, and the bearing of the same ship from station B

Question 1121379: Two Coast Guard stations, A and B, are on an east-west line and are 72 km apart. The bearing of a ship from station A is N 54o E, and the bearing of the same ship from station B is N 31o W.
a) How far is the ship from the east-west line connecting the two Coast Guard stations.
b) How far is the ship from station A?
c) How far is the ship from station B?
Found 2 solutions by josgarithmetic, rothauserc:
Answer by josgarithmetic(39620) (Show Source): You can put this solution on YOUR website!
A, station A
B, station B
S, ship

length AB, 72 kilometers
angle at A, 36 degree
angle at B, 59 degree
angle at S, 85 degree.

Law of Sines will give lengths AS and SB.

and

altitude from S to AB forms two right triangle, sharing this altitude.
Length from S to line AB

or

.
.

Answer by rothauserc(4718) (Show Source): You can put this solution on YOUR website!
the problem states it is giving bearings but really they are directions.
:
The direction to a point is stated as the number of degrees east or west of north or south.
:
The bearing to a point is the angle measured in a clockwise direction from the north line.
:
I am using the information given as directions not bearings
:
:
we need the internal angles of the triangle that joins the ship with stations A and B
:
90 degrees - 54 degrees = 36 degrees and 90 degrees - 31 = 59 degrees and the third angle at the ship is 180 - 36 -59 = 85 degrees
:
now use the law of sines
:
a) sin(36) = distance from ship to AB / 61.9518
:
distance from ship to AB = sin(36) * 61.9518 = 36.4144 km
:
b) 72/sin(85) = distance from A to ship / sin(59)
:
distance from A to ship = 72 * sin(59) / sin(85) = 61.9518 km
:
c) 72/sin(85) = distance from B to ship / sin(36)
:
distance from B to ship = 72 * sin(36) / sin(85) = 42.4822 km
:

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