Question

A forest ranger sights a fire directly to the south. A second ranger, 9 miles east of the first ranger, also sights the fire

The bearing from the second ranger to the fire is S 28° W. How far is the first ranger from the fire?

Answer 1

Let's call the first ranger Ranger A and the second ranger Ranger B. We can draw a diagram of the situation:


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B ----------------- A
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We know that Ranger B is 9 miles east of Ranger A. Let's call the distance from Ranger A to the fire "x" miles. We also know that the bearing from Ranger B to the fire is S 28° W.

A bearing of S 28° W means that the direction from Ranger B to the fire is 28° west of south. We can draw a line showing this direction:


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B -------28°W---- A
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Now we can use trigonometry to find x. We have a right triangle with the following sides:

The side opposite the 28° angle is x miles (the distance from Ranger A to the fire).
The side adjacent to the 28° angle is 9 miles (the distance from Ranger B to Ranger A).
We can use the tangent function to find x:

tan(28°) = x/9

x = 9 tan(28°)

x ≈ 4.62 miles

So the first ranger is approximately 4.62 miles from the fire.