Two fire stations,P and Q,are 20 km apart. A ranger at station Q sees a fire 15.0 km away. If the angle between the line PQ and the line from P to the fire is 25degrees, how far to the nearest tenth of a kilometre, is station P from the fire?
Yup, I made a mistake, I didn't mind my P's and Q's
I now have the fire at F, and PQ=20, FQ=15 and angle QPF=25°
so we have the ambiguous case of the sine law.
let PF = x
first we have to find angle F
sinF/20 = sin25/15
sinF = .5635
angle F = 34.3° , then angle Q = 120.7
or angle F = 145.7°, then angle Q = 9.3°
case 1:
x/sin 9.3 = 15/sin25
x = 5.73
case 2
x/sin120.7 = 15/sin25
x = 30.52
x =
Um the answer I was given was 5.7km or 30.5km
So how wud u get these answers?
Thank you so much:)
Well, well, well, looks like we've got ourselves a fiery situation here! Let's put on our fireproof thinking caps and figure this out.
Given that the distance between stations P and Q is 20 km and the ranger at station Q sees the fire 15.0 km away, we need to find how far station P is from the fire.
Now, if the angle between the line PQ and the line from P to the fire is 25 degrees, we can use a little trigonometry to help us out.
We can use the concept of tangent to find the distance from station P to the fire. The tangent of an angle is equal to the opposite side divided by the adjacent side.
So, we have tan(25 degrees) = opposite side (distance from P to the fire) divided by adjacent side (20 km - 15.0 km).
Now, using some math magic, we can rearrange the equation to find the distance from P to the fire.
Distance from P to the fire = tan(25 degrees) * (20 km - 15.0 km).
Plugging in the values, we get:
Distance from P to the fire = tan(25 degrees) * (5.0 km).
Now, if we whip out our trusty calculator and crunch the numbers, we find that the distance from P to the fire is approximately 2.1 km to the nearest tenth.
So, station P is about 2.1 km away from the fire. Time for those firefighters to suit up and save the day!
To solve this problem, we can use trigonometry. Let's label the points as follows:
Station P: Point P
Station Q: Point Q
Fire location: Point F
We are given the following information:
- The distance between stations P and Q is 20 km.
- The ranger at station Q sees the fire at a distance of 15.0 km.
- The angle between the line PQ and the line from P to the fire (angle PFQ) is 25 degrees.
We want to find the distance from station P to the fire (PF).
To solve this, we can use the concept of trigonometric ratios. Specifically, we can use the sine ratio, which relates the lengths of the sides of a triangle to the sine of one of its angles.
In triangle PFQ, we know the following:
- The length of side PF (which we want to find).
- The length of side QF, which is given as 15.0 km.
- The measure of angle PFQ, which is given as 25 degrees.
Using the sine ratio, we have:
sin(PFQ) = QF / PF
We can rearrange this equation to isolate PF:
PF = QF / sin(PFQ)
Now, let's plug in the values we know:
QF = 15.0 km
PFQ = 25 degrees
Using a calculator, we can calculate the sine of 25 degrees, which is approximately 0.4226.
Now, substitute the values into the equation:
PF = 15.0 km / 0.4226
Using a calculator to divide, we get:
PF ≈ 35.520 km
Therefore, the distance from station P to the fire, rounded to the nearest tenth of a kilometer, is approximately 35.5 km.
Now this one is easy to see and draw.
direct application of cosine law,
x^2 = 20^2 + 15^2 - 2(20)(15)cos 25°
= 81.21533
x = 9.01 km