Answer:
9.91 miles
Step-by-step explanation:
Refer the attached figure
Forest Ranger at point A observes the fire at angle of 41° north of east i.e.∠CAB = 41°
The distance between the two rangers is 15 miles i.e. AB = 15 miles
Forest Ranger at point B observes the fire at at 56° north of west. i.e.∠CBA= 56°
Now we are supposed to find who is closest to the fire
So, we are supposed to find the length of AC and BC
So, first calculate ∠ACB
We will use angle sum property of triangle
Angle sum property of triangle : Sum of all angles of triangle is 180°
So, ∠CBA+∠ACB+∠CAB =180°
56°+∠ACB+41° =180°
97°+∠ACB =180°
∠ACB =180°-97°
∠ACB =83°
Now to find AC and BC we will use law of sines
![\frac{a}{sin A}=\frac{b}{Sin B}=\frac{c}{SinC}](https://tex.z-dn.net/?f=%5Cfrac%7Ba%7D%7Bsin%20A%7D%3D%5Cfrac%7Bb%7D%7BSin%20B%7D%3D%5Cfrac%7Bc%7D%7BSinC%7D)
Refer the attached figure
![\frac{AC}{sin 56}=\frac{BC}{Sin 41}=\frac{15}{Sin83}](https://tex.z-dn.net/?f=%5Cfrac%7BAC%7D%7Bsin%2056%7D%3D%5Cfrac%7BBC%7D%7BSin%2041%7D%3D%5Cfrac%7B15%7D%7BSin83%7D)
So, ![\frac{BC}{Sin 41}=\frac{15}{Sin83}](https://tex.z-dn.net/?f=%5Cfrac%7BBC%7D%7BSin%2041%7D%3D%5Cfrac%7B15%7D%7BSin83%7D)
![BC=\frac{15}{Sin83} \times Sin 41](https://tex.z-dn.net/?f=BC%3D%5Cfrac%7B15%7D%7BSin83%7D%20%5Ctimes%20Sin%2041)
![BC=9.91478](https://tex.z-dn.net/?f=BC%3D9.91478)
![\frac{AC}{sin 56}=\frac{15}{Sin83}](https://tex.z-dn.net/?f=%5Cfrac%7BAC%7D%7Bsin%2056%7D%3D%5Cfrac%7B15%7D%7BSin83%7D)
![AC=\frac{15}{Sin83} \times sin 56](https://tex.z-dn.net/?f=AC%3D%5Cfrac%7B15%7D%7BSin83%7D%20%5Ctimes%20sin%2056)
![AC=12.5289](https://tex.z-dn.net/?f=AC%3D12.5289)
So, BC< AC
So, the ranger who is closest to fire is at a distance of 9.91 miles .
So, Option 1 is true