Two forest fire stations, P and R are 20 km apart. Rajat, the ranger at Station R sees a
fire(F) 15 km. away. Pavit, the ranger at Station P measures the angle between Station R,
himself and the fire, RPF, and finds it to be 21°. It is required that you find how far
Station P is from the fire. Is this an ambiguous case? How do you know? Find the required
distance(s) - to the nearest hundredths. Make sure to include a diagram.
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Yes, it is ambiguous. F lies on a circle of radius 15, with center at R.
There are two tangents from P to the circle.
Find angle PFR using the law of sines.
sinF/20 = sin21°/15
Now, knowing angles P and F, you easily get angle R, and then use either law of sines of law of cosines to find PF.
To determine whether this is an ambiguous case, we need to check if there is only one possible location for Station P given the information provided.
Let's start by creating a diagram to help visualize the scenario:
Station R
(R)
/\
/ \
20 km / ? \ distance x
/ \
/ \
/ F \
(P)------------(F')
In the diagram, R represents the location of Station R, P represents the location of Station P, F represents the observed location of the fire, and F' represents the actual location of the fire.
Given:
- The distance between Stations P and R is 20 km.
- Rajat at Station R sees the fire at a distance of 15 km from Station R.
- The angle between Station R, Station P, and the fire (∠RPF) is 21°.
We need to find the distance from Station P to the fire (∠PFF').
Since we know the angle ∠RPF and the distance from Station R to the fire (15 km), we can use trigonometry to find the distance to the fire (∠PFF').
Using the tangent function, we can write the equation:
tan(∠RPF) = distance to the fire (∠PFF') / distance from Station P to the fire (∠PFP')
Let's solve the equation:
tan(∠RPF) = (∠PFF') / x
tan(21°) = (15 km) / x
x = (15 km) / tan(21°)
x ≈ 42.86 km
So, the required distance from Station P to the fire (∠PFF') is approximately 42.86 km.
Now, let's determine if this is an ambiguous case. To do so, we need to check if there is any other possible location for Station P that satisfies the given information.
If there is only one possible location for Station P, then this is not an ambiguous case. However, if there are multiple possible locations for Station P, then this would be an ambiguous case.
In this scenario, there is only one possible location for Station P that satisfies the distance between Stations P and R, the distance from Station R to the fire, and the angle (∠RPF) measured by Pavit at Station P. Therefore, this is not an ambiguous case.
The required distance from Station P to the fire (∠PFF') is approximately 42.86 km.