Radio direction finders asre set up at points A and B, 8.68 mi apart on an east-west line. From A it is found that the bearing of a signal from a transmitter is N 54.3degE, while from B it is N 35.7degW. Find the distance of the transmitter from B, to the nearest hundredth of a mile.

i got 7.05mi

If you draw this, you have a triangle with ASA. With those two angles, you can get the third angleC.

Use the law of sines:
a/SinA=8.68/sinC
solve for side a.

I agree with your answer.

My diagram has triangle ABC with the interior angle at B equal to 54.3 and the interior angle at A equal to 35.7

(the bearings as given would not be inside the triangle as given)
BTW, angle C = 90

Your answer is what I got for the distance to A,
the distance to B would be 5.064

k thnks

To find the distance of the transmitter from point B, we can use the concept of triangulation. Triangulation is a method that uses the angles from two reference points to calculate the distance to an unknown point.

1. Draw a diagram: Start by drawing a diagram with points A and B, 8.68 miles apart on an east-west line. Label the transmitter as point T.

2. Determine the angles: From point A, the bearing of the signal from the transmitter is N 54.3° E.
From point B, the bearing of the signal from the transmitter is N 35.7° W.

3. Calculate the missing angles: In order to calculate the distance of the transmitter from point B, we need to find the two missing angles.

a. Angle A: Since the given bearing from point A is N 54.3° E, we subtract 90° (north) from it to find the angle A:
A = 54.3° - 90° = -35.7°

b. Angle B: Similarly, since the given bearing from point B is N 35.7° W, we can subtract 90° from it to find the angle B:
B = 180° - 35.7° = 144.3°

4. Apply the Law of Sines: The Law of Sines states that for any triangle, the ratio of the length of a side to the sine of its opposite angle is constant. Using this law, we can set up an equation to solve for the distance CT (where C is the vertex opposite to side CT).

CT / sin(A) = AB / sin(B)
CT / sin(-35.7°) = 8.68 / sin(144.3°)

5. Solve for CT: Rearrange the equation to solve for CT.
CT = (8.68 * sin(-35.7°)) / sin(144.3°)

6. Calculate the distance: Plug the values into a calculator to find the distance CT.

CT = (8.68 * sin(-35.7°)) / sin(144.3°) ≈ 7.05 miles

Therefore, the distance of the transmitter from point B is approximately 7.05 miles to the nearest hundredth of a mile, confirming your result.