Question

A forest ranger at lookout a sights a fire directly north of her position. Another ranger at lookout b, exactly 2 kilometers directly west of a, sights the same fire at a bearing of n41.2°e. How far is the fire from lookout a? Round your answer to the nearest 0.01 km?

Answer 1

Using trigonometric ratios, the distance of the fire from lookout A is found to be approximately 2.69 kilometers when sighted at a bearing of N41.2°E from lookout B, which is exactly 2 kilometers west of A.

Step-by-step explanation:

To determine how far the fire is from lookout A when a forest ranger at lookout B, which is 2 kilometers west of A, sights the same fire at a bearing of N41.2°E, we can use trigonometric methods. We can imagine a right-angled triangle where one leg (the distance west from A to B) is known, and the angle at lookout B is known (41.2° from the north towards the east). We are looking for the hypotenuse of this triangle, which will be the distance from lookout A to the fire.

Let's represent the distance of the fire from lookout A as 'x.' To find 'x,' we apply the law of cosines or use trigonometric ratios. In this case, since we have a right-angled triangle and we know the angle at B, we can use the cosine function:

cos(41.2°) = Adjacent / Hypotenuse

cos(41.2°) = 2 km / x

Solving for 'x' gives us:

x = 2 km / cos(41.2°)

Calculating this value, we find that the distance from lookout A to the fire is approximately:

x ≈ 2.69 km (rounded to the nearest 0.01 km)

The fire is therefore approximately 2.69 kilometers away from lookout A.