Geometry Problem, I don't even know how to start!

When I did the solution with your first method for the second part of the question i found that i only got the one, wrong answer.

CosPBC = 30/40
PBC= cos-1 (30/40)
PBC = 41.41
60- 41.41 = 18.59
angle CBP = 18.59
Sin18.59 = ABP/60

connects A's pipeline at 19.13 along. what have i done wrong?

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error is in Sin18.59 = ABP/60. Triangle ABP not rightangled;use sine rule: x/sin18.59 = 40/sin30; x=80*sin18.59 = 25.5. Find other position by AC=51.96, CP =51.96-25.5=26.46;51.96+26.46. Allow email so we can communicate better! L.E.Grant solution good too:sinx=3/4 so angleAPB=48.6 or 180-48.6=131.4

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Start by drawing yourself a diagram. You have a good idea where the towns are relative to each other and where the existing pipeline is. Then draw a circle for the "40km" pipe that needs to be built. Where this crosses the existing pipeline gives you the answer(s) you need

Most of the time, a diagram for this kind of problem is the best place to start.

N60E is at 30 degrees to the road.
Let C be one point 40 kn from town B
then BC/sin(30) = AB/ sin(x) where x is the angle at C
AB = 60km
BC = 40 km
so sin(x) = 60sin(30)/40 = 3sin(30)/2 = 3/4
You can look up the angle(s), and then use these to find AC