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 β 8 * 0.8387
BF β 6.71 miles
AF = BF * sin(33 degrees)
AF β 6.71 * 0.5446
AF β 3.65 miles
Therefore, the first ranger is approximately 3.65 miles away from the fire.
\boxed{AF \approx 3.65 \text{ miles}}