Distances at Sea---Solving Right Triangles

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greatwhiteshark

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The navigator of a ship at sea spots two lighthouses that she recognizes to be 3 miles apart. She then determines that the angles formed between the two line-of-sight observations of the lighthouses and the line from the ship directly to shore are 15 degrees and 35 degrees.

a) How far is the ship from shore?
b) How far is the ship from lighthouse A and lighthouse B?

MY WORK:

tan15 degrees = 3 miles/side a

side a = 3 miles/tan15 degrees

a = 11.2 miles (Distance from ship to Lighthouse A) or so I thought.

I found the distance from the ship to Lighthouse B to be 4.3 miles.

BOTH answers are wrong. I was not able to find the distance from the ship to shore. Help. What am I doing wrong?

The book's answer:

Distance from ship to shore: 3.1 miles
Distance from ship to lighthouse A: 3.2 miles
Distance from ship to lighthouse B: 3.8 miles.

MY QUESTION:

How can I find these answers?
 
You need a better picture on this one. 3 miles is NOT the side of a Right Triangle. This makes your first equation rather suspect.
 
I think we also need to know if the ship is BETWEEN the light houses or are they both on the same side?
 
OK, let's try this. First, we need a CLEAR definition.

Draw a piece of straight shoreline.
One one end, label Point A, the location of one lighthouse.
On the other end, label Point B, the location of the other lighthouse.
BETWEEN them, label Point S, the point from the boat nearest the shore.

Perpendicular to the shoreline, at Point S, draw a perpendicular.
Out on the perpendicular, label Point T, the location of the ship.

We are given:
Segment AB is '3 miles'
Angle TSA is 15º
Angle TSB is 35º

We are NOT given (so, just name them for now):
Segment AS is 'a miles'
Segment SB is 'b miles'
Then a+b = 3, so Segment SA is '(3-a) miles'
Segment TS is 's miles'

Observe Trigonometric Relationships:
tan(15º) = a/s
tan(35º) = (3-a)/s

These can be solved for 'a' and 's'.
'b' easily follows.
The pythagorean theorem will handle the line-of-sight distances for TB and TA.

Note: This is based on the assumption that the ship is between the lighthouses. If the lighthouses are on the same side, the distance to shore is 6.94 miles.
 
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