Question

a forest ranger sights a fire directly to the south. a second ranger, 6 miles east of the first ranger, also sights the fire. the bearing from the second ranger to the fire is s 30° 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 distance between the two rangers is 6 miles. The angle at the second - ranger's position with respect to the line connecting the two rangers and the line from the second ranger to the fire is $30^{\circ}$.

Step2: Use tangent function

We know that $\tan30^{\circ}=\frac{6}{x}$. Since $\tan30^{\circ}=\frac{1}{\sqrt{3}}$, we have the equation $\frac{1}{\sqrt{3}}=\frac{6}{x}$.

Step3: Solve for $x$

Cross - multiply to get $x = 6\sqrt{3}$.

Step4: Approximate the value

$x\approx6\times1.732 = 10.4$ (rounded to the nearest tenth).

Answer:

$10.4$