Points A and B are separated by a lake. To find the distance between them, a surveyor locates a point C on land such than ∠ C A B = 48.3 ° . Find the distance across the lake from A to B.

Question

How can I find this answer if B is 487m and A is 347m?

Answer 1

Another variation of the problem interpretation follows…

Let us assume that “A=347m” and “B=487m” is in error should have been written “a=347m (side opposite ∠A)” and “b= 487m (side opposite ∠B).”  Let us also assume that ∠CAB=48.3° IS CORRECTLY identified as opposite side “a.”

Then, as Mark suggests, the Law of Sines can be applied.  This would yield a result of…

            a = 347m = mBC

            b = 487m = mAC

       SIN∠B = 0.75(487)/347

       SIN∠B = 1.05

SIN⁻¹ (1.05) = ∠B

        ∴∠B = Ø…domain error, no solution

So, now let us assume that “a and b” should have been reversed…

             a = 487m = mAC

             b = 347m = mBC

Again, applying the Law of Sines yields…

SIN 48.3°/487 = SIN(∠B)/347

     0.747(347) = 487(SIN∠B)

         259.08 = 487(SIN∠B)

         SIN∠B = 0.532

   SIN⁻¹ (0.532) = ∠B

           ∴∠B = 32.14° 

           ∴∠C = 99.56°

Setting up another Law of Sine ratio equality, we can solve for “c”…

       SIN(∠A)/a= SIN(∠C)/c

 SIN 48.3°/487= SIN 99.56°/c

        0.747(c) = 0.986(487)

        0.747(c) = 480.23

            ∴ c = mAB = 643.20m…a reasonable result

Let us check our solution values comparing the relationship values of the Law of Sines…

1/[(SIN∠C) / c] = 1/[(SIN99.56°)/643.20] = 652.25

1/[(SIN∠B) / b] = 1/[(SIN32.14°)/347]      = 652.27

1/[(SIN∠A) / a] = 1/[(SIN48.3°)/487]       = 652.26

…all ratio values can be considered equal according to the Law of Sines.  Our calculated values are accurate.  Therefore, the measure of segment AB is 643.20 meters across the lake.

Answer 2

So, let's start with what we know

We have side A being 347m, side B being 487m, and i'm a little confused on where the angle is supposed to fall. Standard notation for angle writing is, for example ABC, would be angle B which is between the lines A and C. Here, it looks as if we were given angle A between lines B and C, but I don't think that makes sense in the context of the problem. I'm assuming that the angle is meant to be angle C, between sides A and B. The reason I am assuming this is because when you have two side lengths and the angle between them, you are able to use the law of cosines to solve the triangle. So, moving forward withg the problem, I will solve it under this assumption, however if you read this and I am incorrect I'm happy to adjust for the errors.

So, we have two sides, a=347m, b=487m, and the angle between them, C=48.3 degrees. The side, c, connects points A and B, and by using the law of cosines to solve for c, we will find the distance across the lake between points A and B.

The law of cosines is as follows:

a²+b²-2abCos(C)=c²

Now, we just insert

(347)²+(487)²-2(347)(487)Cos(48.3)=

120409+237169-337978(0.6652)=132744.7752

√(132744.7752)=364.34m=length of side c=distance between points A and B.

I hope that this explaination is able to make things clear! Let me know if I can do anything else to help. Good luck!