Two fire rangers in watch towers that are 10 miles apart spotted a fire at the same time. the first ranger indicated that the position of the fire made an angle of 39° with the line between the towers. the second ranger indicated that it made an angle of 34° with the same line. how far is the second tower from the fire?

5.455 miles

Step-by-step explanation:

Suppose the height of watch towers are same(h)

and distance between them is 10 miles

first ranger indicated angle

second ranger indicated angle

let x be the distance from second ranger to fire

thus

equating h values

10tan39=1.48429x

x=5.444  miles

(sin 42)/31. = sinx)/37

sinx = 37 (sin42)/31

= 37× 0.66/31 = 0.787

x= 51.9

other angle opposite to y = 180-51.9-42

= 180- 93.9 =86.1

so

sin42)/31 = sin 86.1)/y

0.021 = 0.997 /y

y= 0.997/0.021 =47.47