Question

a forest ranger sights a fire directly to the south. a second ranger, 5 miles east of the first ranger, also sights the fire. the bearing from the second ranger to the fire is s 32° w. how far is the first ranger from the fire? how far is the first ranger from the fire? mi (round to the nearest tenth of a mile.)

Explanation:

Step1: Identify the right - triangle

Let the first ranger be at point $A$, the second ranger be at point $B$ which is 5 miles east of $A$, and the fire be at point $C$. The bearing from the second ranger to the fire is $S32^{\circ}W$, so the angle $\angle ABC = 32^{\circ}$ in the right - triangle $ABC$ with right - angle at $A$.

Step2: Use the tangent function

We know that in a right - triangle $\tan\theta=\frac{\text{opposite}}{\text{adjacent}}$. Here, $\theta = 32^{\circ}$, the adjacent side to $\angle ABC$ is $AB = 5$ miles, and the opposite side is $AC$ (the distance from the first ranger to the fire). So $\tan(32^{\circ})=\frac{AC}{5}$.

Step3: Solve for $AC$

We can rewrite the equation as $AC = 5\times\tan(32^{\circ})$. Since $\tan(32^{\circ})\approx0.6249$, then $AC = 5\times0.6249 = 3.1245\approx3.1$ miles.

Answer:

$3.1$