SOLUTION: Could you please help me with this problem about Law of Sines? I understand this is an Ambiguous case but I am getting an incorrect answer. Thank you in advance! <a rel=nofollo

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Question 1198037: Could you please help me with this problem about Law of Sines? I understand this is an Ambiguous case but I am getting an incorrect answer. Thank you in advance!
IMG-6246
Found 2 solutions by josgarithmetic, MathTherapy:
Answer by josgarithmetic(39621) About Me  (Show Source):
You can put this solution on YOUR website!
just Law Of Sines?
The upper ship, U
the lower ship, B

system%28sin%2845%29%2F800=sin%28U%29%2Fd=sin%28B%29%2F1000%2C45%2BU%2BB=180%29

Look at that first combination equation which uses Law Of Sines.
Transitive Property means that sin%2845%29%2F800=sin%28B%29%2F1000, so this should help to find a value or values for angle at point B.

That seems to give without much complication, B=62%2Adegrees and U=73%2Adegrees. Returning to Law Of Sines again to find d should be very clear to do.
.
.
I'm finding d=1082.


** solution is incomplete.

Answer by MathTherapy(10555) About Me  (Show Source):
You can put this solution on YOUR website!
Could you please help me with this problem about Law of Sines? I understand this is an Ambiguous case but I am getting an incorrect answer. Thank you in advance!
.
I wonder what incorrect answer you're getting.

Let the 1st boat's location be D, the 2nd boat's, C, and yours, B
Then DC = 800m = b, and BD = 1,000m = c
We now apply the Law of Sines, matrix%281%2C3%2C+sin+%28B%29%2Fb%2C+%22=%22%2C+sin+%28C%29%2Fc%29 to get: matrix%281%2C3%2C+sin+%2845%5Eo%29%2F800%2C+%22=%22%2C+sin+%28C%29%2F%221%2C000%22%29
                                                   800 * sin (C) = 1,000 * sin (45o) ----- Cross-multiplying
                                                          
                                                              

With ∡s B and C being 45o and 62o, respectively, ∡D will be: 180 - (45 + 62) = 180 - 107 = 73o
You need to know how far the 2nd boat is from you, which is BC or d

Again, apply the Law of Sines, matrix%281%2C3%2C+sin+%28D%29%2Fd%2C+%22=%22%2C+sin+%28B%29%2Fb%29 to get: matrix%281%2C3%2C+sin+%2873%5Eo%29%2Fd%2C+%22=%22%2C+sin+%2845%5Eo%29%2F800%29
                                                    d * sin (45o) = 800 * sin (73o) ----- Cross-multiplying
                                                              

As seen, ∡C is the MIDDLE angle (62o), compared to 45o and 73o for ∡s B and D (the largest), respectively.
However, ∡C can also be 118o (180o - 62o), thereby making it the largest angle, instead of D. 

With ∡C being 118o, ∡B remaining at 45o (always will), ∡D will now be: 180 - (118 + 45) = 180 - 163 = 17o.
This makes ∡D the smallest, which in turn makes d, or BC the shortest of the 3 sides, at 330.7808241 ≈ 330.78, or approximately 331 m.

So, unless you receive some clue (the 2nd boat is CLOSER to you than it is to the 1st boat, or the 2nd boat is FARTHER away from you 
than it is to the 1st boat) that'll indicate which of these 2 sides "d" could be, its TRUE distance from you cannot be determined

I forgot to check the 2 possibilities usng the TRIANGLE INEQUALITY THEOREM, hence the crossout above. The theorem states that ANY side
of a triangle will be less than the sum of the triangle's other two sides.

Doing that for the 1st scenario (distance or length of 1,082), and using 1,082 as one of the triangle's sides, we determine if this is 
TRUE: 1,082 < 1,000 + 800. We clearly see that this is FALSE, so 1,082 m CANNOT be one of the sides of this triangle. Hence, such a triangle 
is NON-EXISTENT, and you CANNOT be 1,082 m from the 2nd boat. 

We then try 331 m as one of the triangle's sides to determine if the following is TRUE: 331 < 1,000 + 800. We clearly see that this is
TRUE, as 331 is definitely less than 1,000 + 800 = 1,800. 
Therefore, such a triangle DOES EXIST and so, the 2nd boat is approximately 331 m from you.