Answer
about 136 miles
Work Step by Step
1. Name the intersection as point A.
At A draw a vertical north/south line and a horizontal east/west line.
Place a point B south of A, distance from A = 2(55)=110 miles.
2. Place point C west of of A, such that the bearing from B to C is 32$4^{0}.$
(point C will be $360-324=36^{o}$ west of north, when observing from B)
3. Note that we have a right triangle. Use CAH in SOHCAHTOA.
$\displaystyle \cos 36^{o}=\frac{110}{d}$
$d\cos 36^{o}=110$
$ d=\displaystyle \frac{110}{\cos 36^{o}}\approx$135.967477525
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