Question

A forest ranger looking out from a ranger’s station can see a forest fire at a 35 º angle of depression. The ranger’s position is 100 feet above the ground. How far is it from the ranger’s station to the fire? Round to the nearest tenth,

Answer 1

To find the distance from the ranger's station to the fire, we use trigonometry. The situation can be modeled as a right triangle where:

- The height of the ranger's station (100 feet) is the opposite side.
- The distance from the ranger's station to the fire is the hypotenuse.
- The angle of depression is 35 degrees.

Since the angle of depression from the ranger's station corresponds to an angle of elevation from the fire level to the ranger's station, we can consider the angle of elevation (35 degrees) in our calculations.

Using the sine function, which relates the opposite side to the hypotenuse, we have:

\[
\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}
\]

where:
- \(\theta = 35^\circ\)
- \(\text{opposite} = 100\) feet (height of the ranger's station)
- \(\text{hypotenuse} = d\) (distance from the ranger's station to the fire)

So we can write:

\[
\sin(35^\circ) = \frac{100}{d}
\]

To find \(d\), we rearrange this equation:

\[
d = \frac{100}{\sin(35^\circ)}
\]

Now we calculate \(\sin(35^\circ)\):

\(\sin(35^\circ) \approx 0.5736\)

Thus, we can find \(d\):

\[
d = \frac{100}{0.5736} \approx 174.0
\]

Therefore, the distance from the ranger's station to the fire is approximately **174.0 feet**.

Rounding to the nearest tenth, the answer is **174.0 feet**.