Question

Two forest-fire towers, A and B, are 20.3 km apart. From tower 4, the compass heading for tower B is $80°E. The ranger in each tower sees the same forest fire. The heading of the fire from tower A is N50°E.
The heading of the fire from tower B is N60°W. How far, to the nearest tenth of a kilometre, is the fire from each tower?

Answer 1

To find the distance from each tower to the fire, we can use the Law of Cosines. Let's label the distances from tower A and B to the fire as x and y, respectively.

For tower A:
By using the Law of Cosines, we have:
x^2 = 20.3^2 + d^2 - 2*20.3*d*cos(50°)
x^2 = 412.09 + d^2 - 40.6d*cos(50°)

For tower B:
By using the Law of Cosines, we have:
y^2 = 20.3^2 + d^2 - 2*20.3*d*cos(120°)
y^2 = 412.09 + d^2 + 40.6d*cos(60°)

Now, we have a system of equations:
x^2 = 412.09 + d^2 - 40.6d*cos(50°) (1)
y^2 = 412.09 + d^2 + 40.6d*cos(60°) (2)

By solving equation (1) and equation (2), we can find the values of x and y. Then, we can calculate the distances from each tower to the fire.