Posted by **Nikki** on Saturday, November 26, 2011 at 1:12am.

two forest fire stations,P and Q are 20.0 km apart, A ranger at station Q sees a fire 15.0 km away, if the angle between line PQ and the line form P to the fire is 25 , how far to the nearest tenth of a kilometre is statin P from the fire

- Math -
**Steve**, Saturday, November 26, 2011 at 5:02am
Not enough information.

Let the fire be at point F.

If the angle at P were 36.9 degrees, then we would have a right triangle with legs 15 and 20, and hypotenuse PF = 25.

However, since angle P is less than 36.9 degrees, a circle of radius 15 will cut the line PF extended in two places, either one of which would be a solution. The angle at F could be either acute or obtuse.

However, using the law of sines,

15/sin15° = 35.5 = 20/sinF

sinF = 20/35.5

F = 34° or 146°

PF/sin(180-(25+34)) = 35.5 makes PF = 30.42

PF/sin(180-(25+146)) = 35.5 makes PF = 5.38

SO, the distance PF depends on which direction the ranger was looking when he saw the fire.

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