Answer
1.93 miles
Work Step by Step
The triangle formed between stations A and B and the Receiver R will be ∆ABR. We know the line AB to be 3.46 miles long, so we have one side of the triangle and are indirectly given two of its angles, A and B.
Find angle A by subtracting the 47.7º from the right angle between the East and North.
$90º - 47.7º=42.3$
Find angle $B$ by subtracting 270º from the given 302.5º.
$302.5º-270º=32.5º$
The remaining angle R can be found because the sum of a triangle's interior angles must be 180º.
So,
$R+42.3º+32.5º=180º$
Solve for R
$R = 180º-42.3º-32.5º$
$R = 105.2º$
Now use the law of sines to relate angle R and line AB with line AR and its opposite, angle B.
The law of sines relates the length of a triangle's sides with the sine of their opposite angle, such that:
$\frac{AB}{\sin R} = \frac{AR}{\sin B}$
Solve for AR
$ {AR}= \frac{{AB}\times \sin B}{\sin R}$
Plug known values in.
$ {AR}= \frac{{3.46}\times \sin 32.5}{\sin 105.2}$
Solve with a calculator
$AR ≈ 1.93$
The distance between point A and point R is about 1.93 miles
