Bearings Application Problem

Question

A boat leaves the harbor entrance and travels in the direction N 29 ∘ E. After 19 miles, the captain realizes he's made a navigation error, and changes direction to S 59 ∘ E. After traveling 11 miles at this bearing, how far is the boat from the harbor entrance, and what is the bearing of the boat from the harbor entrance?

Answer 1

A sketch would look like this:

Angle α is the alternate interior angle match to the 29° angle therefore α = 29° making the whole angle at the top of the triangle 29° + 59° = 88°

We can use the law of cosines to solve for side x:

x² = 19² + 11² - 2(19)(11)cos(88°)

x² = 467.412

x = √467.412 = 21.62 miles

Now we can use the law of sines to solve for angle θ:

sin(88°)/21.62 = sin(θ)/11

11sin(88°) = 21,62sin(θ)

sin(θ) = 11sin(88°)/21,62 = 0.5085

θ = sin^-1(0.5085) = 30.6°

Adding that to the 29° we get a total angle of 59.6° off of north

So the distance the boat is from the harbor is 21.6 miles and the bearing then is N 59.6° E