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a forest ranger sights a fire to the south a second ranger 8 miles east of the first ranger also sights the fire the beari

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A forest ranger sights a fire to the South. A second ranger, 8 miles East of the first ranger, also sights the fire. The bearing from the second ranger to the fire is S 33(degrees). How far is the first ranger from the fire?

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Answer to a math question A forest ranger sights a fire to the South. A second ranger, 8 miles East of the first ranger, also sights the fire. The bearing from the second ranger to the fire is S 33(degrees). How far is the first ranger from the fire?

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Let's denote the position of the first ranger as point A, the position of the fire as point F, and the position of the second ranger as point B.

Given:
AB = 8 miles
Angle FBA = 33 degrees

To find the distance between the first ranger (A) and the fire (F), denoted as AF, we can use trigonometry.

We first find the length of side BF using the cosine rule:
cos(33 degrees) = BF / AB
BF = AB * cos(33 degrees)
BF = 8 * cos(33 degrees)

Next, we can use the sine rule to find the distance AF:
sin(33 degrees) = AF / BF
AF = BF * sin(33 degrees)

Calculating the values:
BF = 8 * cos(33 degrees)
BF ≈ 8 * 0.8387
BF ≈ 6.71 miles

AF = BF * sin(33 degrees)
AF ≈ 6.71 * 0.5446
AF ≈ 3.65 miles

Therefore, the first ranger is approximately 3.65 miles away from the fire.

\boxed{AF \approx 3.65 \text{ miles}}

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