solve problem

Question

An island is located is located 36 miles N 32°39' W of a city. A freighter in distress, radio's it's position as N 11° 24' of the island and N 16° 28' W of the city. How far is the freighter from the city.

Answer 1

I'll solve this navigation problem using the Law of Sines.

Step 1: Set up the triangle

Let C = city, I = island, F = freighter

Given information:

Island is 36 miles from city at bearing N 32°39' W

Freighter's bearing from island: N 11°24' E

Freighter's bearing from city: N 16°28' W

Step 2: Find the angles of the triangle

At the city (angle C):

The angle between the two bearings from C is:

32°39' - 16°28' = 16°11'

At the island (angle I):

From I, the bearing to C is S 32°39' E (opposite of N 32°39' W)

The bearing to F is N 11°24' E

Angle between these bearings:

32°39' + 11°24' = 44°03'

At the freighter (angle F):

Sum of angles = 180°

F = 180° - 16°11' - 44°03' = 119°46'

Step 3: Apply Law of Sines

We know: CI = 36 miles (side opposite angle F)

Law of Sines: CF/sin(I) = CI/sin(F)

CF/sin(44°03') = 36/sin(119°46')

CF = 36 × sin(44°03')/sin(119°46')

CF = 36 × 0.6947/0.8660

CF = 36 × 0.8022

CF ≈ 28.9 miles

The freighter is approximately 29 miles from the city.