SOLUTION: A ship leaves a dock at 13 mph. After an hour it is seen from a lighthouse 12 miles east of the dock to bearing N12°30’W. How far is it from the lighthouse? What is its bearing

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Question 1194112: A ship leaves a dock at 13 mph. After an hour it is seen from a lighthouse 12 miles east of the dock to bearing N12°30’W. How far is it from the lighthouse? What is its bearing from the dock?
Found 2 solutions by Edwin McCravy, math_tutor2020:
Answer by Edwin McCravy(20056)   (Show Source): You can put this solution on YOUR website!
 

Let the dock be at D, the lighthouse at L, and the ship at S.

A bearing of N12o30'W means this: 

Think of starting by putting your elbow at L. Point your arm North (up), 

then swing your arm toward the West (left) through an angle of 12o30'.  

Since 30' means half a degree, I will write the angle as 12.5o. 

To find ∠DLS we subtract 90o-12.5o = 77.5o.

Since the ship travels at 13 mph, and travels 1 hour, the distance 

DS = 13 miles, the distance from the dock.   

We find ∠S by using the law of sines:







∠S = 64.3157657o

Now we find ∠LDS by using the fact that the three angles of
ΔDLS must have sum 180o.

∠LDS = 180o - 64.3157657o - 77.5o = 38.1842343o

To find the distance from the lighthouse to the ship, LS, we use the
law of cosines:





miles.

That's the first thing you were asked for.

-------------

To get the bearing angle from D, we subtract ∠LDS from 90o

90o - 38.1842343o = 51.8157657o

For the bearing from D, think of starting by putting your elbow at D. 
Point your arm North (up), then swing your arm toward the East (right) 
through an angle of 51.8157657o.

Since the first bearing was given in degrees and minutes, we change
the decimal part to minutes by multiplying by 60':

(0.8157657)(60') = 48.945942' which we round to 49'

So the bearing from D is N51o49'E.

Edwin
Answer by math_tutor2020(3817)   (Show Source): You can put this solution on YOUR website!

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

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: 8.23 miles (approximate)

Question: What is its bearing from the dock?
Answer: N51°49’E (approximate)
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