Question 1194112
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In common practice, nautical miles and knots (nautical miles per hour) are used when it comes to ocean environments; however, I'll stick to miles and mph to keep with the given context.


Diagram from a bird's-eye view
<img src = "https://i.imgur.com/zeasBNW.png">
The points of interest
D = dock
B = boat
H = lighthouse
M = point directly north of D
N = point directly north of H


point H is 12 miles directly east of D, hence segment DH = 12
The boat travels at a speed of 13 mph, and does so for 1 hour, so it is 13*1 = 13 miles from the dock. 
We assume that the boat travels in a straight line.
Segment DB = 13.


The bearing N12°30’W means we start looking directly north and turn 12°30’ to the west.
Recall that 60 arcminutes is equal to 1 full degree.
Therefore, 30 arcminutes is 1/2 = 0.5 of a degree.
In short, 12°30’ = 12.5°
This is the angle in red, which is angle NHB.


The blue angle is 90 - 12.5 = 77.5 degrees
For triangle BDH, this is interior angle H.


Let x be the measure of angle B
Use the Law of Sines to determine x
sin(B)/b = sin(H)/h
sin(x)/12 = sin(77.5)/13
sin(x) = 12*sin(77.5)/13
sin(x) = 0.90119631426456
x = arcsin(0.90119631426456)
x = 64.3157656946611
which is approximate.


Then we can say
B+D+H = 180
x+y+77.5 = 180
64.3157656946611+y+77.5 = 180
y+141.815765694661 = 180
y = 180-141.815765694661
y = 38.184234305339
which is the approximate measure of angle BDH in purple


From there,
angle MDB = 90 - (angle BDH)
angle MDB = 90 - y
angle MDB = 90 - 38.184234305339
angle MDB = 51.815765694661
which is also approximate.


We have 51 full degrees, plus an extra 0.815765694661 of a degree.
Multiply this with 60 to convert it to arcminutes
60*0.815765694661 = 48.94594167966
That rounds to about 49 arcminutes
We can think of angle MDB as 51°49’ approximately, which produces the bearing N51°49’E


Here's how to interpret that bearing: Place yourself at point D and look directly north to point M. Then turn roughly 51°49’to the east so you can spot the boat (B)


All of this addresses the bearing of the boat from the dock. 
Let's now find the value of d (aka side length HB) to determine how far it is from the lighthouse H to the boat B.


Use the law of sines again
sin(H)/h = sin(D)/d
sin(77.5)/13 = sin(38.184234305339)/d
d*sin(77.5) = 13*sin(38.184234305339)
d = 13*sin(38.184234305339)/sin(77.5)
d = 8.2316199904907
The distance is about 8.23 miles. 


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Summary:


Question: How far is it from the lighthouse?
Answer: <font color=red>8.23 miles</font> (approximate)


Question: What is its bearing from the dock?
Answer: <font color=red>N51°49’E</font> (approximate)
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