Using Pythagoras theorem and trigonometric rule, the ranger who is closest to the fire is 10.18 miles.
Trigonometry deals with the relationship between the sides and angles of a right-angle triangle.
Given
A forest ranger in the west observation tower spots a fire 41° north of the east.
Fifteen miles directly east, the forest ranger in the east tower spots the same fire at 56° north of west.
Let AC be the x and BC will be 15 -x.
And FC is the h.
In ΔACF.
[tex]\rm h = x tan 41\\\\h = 0.8693 \ x[/tex]...1
In ΔBCF.
[tex]\rm h = (15-x) tan 34\\\\h = 10.1176 - 0.6745x[/tex]...2
From equation 1 and 2
[tex]\rm 0.8693 \ x = 10.1176 -0.6745x[/tex]
On simplifying, we have
x = 6.554
Then 15-x will be 8.446
And h will be 5.697
Then by Pythagoras theorem, the ranger who is closest to the fire will be
FB² = 8.446² + 5.697²
FB = 10.18 miles
The diagram is shown below.
Thus, the ranger who is closest to the fire is 10.18 miles.
More about the trigonometric link is given below.
https://brainly.com/question/22698523
