Two fire towers, A and B, are 20.3 km apart. From tower A, the compass heading for tower B is S80E. The ranger in each tower sees the same forest fire. The heading of the fire from tower A is N50E. The heading of the fire from tower B is N60W. How far, to the nearest tenth of a kilometre is the fire from each tower?

The answer is 7.4 and 16.5 but I don't know how to get it??

see Damon's solution in the related questions below.

To find the distance from each tower to the fire, we can use trigonometry and the given compass headings.

Let's start with tower A. The compass heading from tower A to tower B is S80E, which means the angle formed between the line connecting tower A and tower B, and the north direction is 80 degrees (since it is south of east).

Now, let's consider the fire's heading from tower A, which is N50E. This means the angle formed between the line connecting tower A with the fire and the north direction is 50 degrees (since it is north of east).

We can use these two angles, along with the given distance between the two towers (20.3 km), to find the distance from tower A to the fire using the law of sines.

Using the law of sines, we have the following equation:

sin(A) / a = sin(B) / b

Where A and B are the angles opposite sides a and b, respectively.

Let's denote the distance from tower A to the fire as x, and the distance from tower B to the fire as y.

For tower A:
Angle A = 180 - (90 - 50) = 140 degrees
Angle B = 180 - (90 + 80) = 10 degrees

Using the law of sines, we have:

sin(140) / x = sin(10) / 20.3

Rearranging and solving for x:

x = (sin(140) * 20.3) / sin(10)
x ≈ 7.4 km (rounded to the nearest tenth)

Similarly, we can calculate the distance from tower B to the fire as follows:

For tower B:
Angle A = 180 - (90 + 60) = 30 degrees
Angle B = 180 - (90 - 50) = 140 degrees

Using the law of sines again:

sin(30) / y = sin(140) / 20.3

Rearranging and solving for y:

y = (sin(30) * 20.3) / sin(140)
y ≈ 16.5 km (rounded to the nearest tenth)

Therefore, the fire is approximately 7.4 km from tower A and 16.5 km from tower B.