Two forest fire towers, A and B, are 20.3 km apart. From tower A, the bearing of tower B is 70 degrees. The ranger in each tower observes a fire and radios the bearing of the fire form the tower. The bearing from tower A is 25 degrees and from tower B is 345 degrees. How far, to the nearest tenth of a kilometre, is the fire from each tower?
I know how to use sine and cosine laws, just not how to draw it to start the question. Thanks!
To solve this problem, we can start by drawing a diagram. Let's label the two towers as A and B, and the location of the fire as F. From the given information, we know that the distance between towers A and B is 20.3 km.
Next, we need to determine the angles at each tower. The bearing from tower A to B is given as 70 degrees, so we can draw a line from A to B with an angle of 70 degrees.
A(25 degrees)
_____|_____
\ /
\ /
\ /
\ /
\ /
F
|
|
B(345 degrees)
We are also given that the bearing of the fire from tower A is 25 degrees and from tower B is 345 degrees. This means we can draw lines from towers A and B to the fire location F with angles of 25 degrees and 345 degrees, respectively.
Since we are given the angles and the distance between the towers, we can use the cosine law to find the distance from each tower to the fire. The cosine law states that in a triangle with sides a, b, and c, and the angle opposite side c is C, the following formula can be used:
c^2 = a^2 + b^2 - 2ab * cos(C)
In this case, we want to find the distances from each tower to the fire, so let's call those distances dA and dB. The formula for each distance is:
dA^2 = 20.3^2 + x^2 - 2 * 20.3 * x * cos(25)
dB^2 = 20.3^2 + x^2 - 2 * 20.3 * x * cos(345)
where x is the distance from tower A to the fire, and dA and dB are the distances we want to find.
Using the given information, we can plug in the values into the formulas and solve them to find the distances.
To solve this problem, you will need to draw a diagram to visualize the given information. Here are the steps to draw the diagram:
1. Draw two points on a piece of paper to represent the forest fire towers A and B.
2. Label the distance between the two towers as 20.3 km.
3. From tower A, draw a line segment in the direction of the bearing of tower B. This line segment will form an angle of 70 degrees with the horizontal line.
4. From tower B, draw a line segment in the opposite direction of the bearing of tower A. This line segment will form an angle of 25 degrees with the horizontal line.
5. Label the angles formed at tower A and tower B with their respective values (70 degrees and 25 degrees).
6. Finally, label the distance from each tower to the fire as variables (let's call them 'a' and 'b' respectively).
Once your diagram is complete, you can use the sine or cosine laws to solve for 'a' and 'b' and find the distances from each tower to the fire.