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: Set up the right - triangle

Let the distance between the first ranger and the fire be $x$ miles. The horizontal distance between the two rangers is 5 miles. The angle formed at the second ranger with respect to the line connecting the two rangers and the line from the second ranger to the fire is $32^{\circ}$.

Step2: Use the tangent function

We know that in a right - triangle, $\tan\theta=\frac{\text{opposite}}{\text{adjacent}}$. Here, $\tan(32^{\circ})=\frac{x}{5}$.

Step3: Solve for $x$

Multiply both sides of the equation $\tan(32^{\circ})=\frac{x}{5}$ by 5. So, $x = 5\times\tan(32^{\circ})$.
Since $\tan(32^{\circ})\approx0.6249$, then $x=5\times0.6249 = 3.1245\approx3.1$.

Answer:

$3.1$