From a lookout tower on a mountain, a forest ranger spots a fire in a nearby valley at an angle of depression of 26%. The ranger is 4500 ft above sea level and the valley is 2000 ft above sea level. What is the horizontal distance from the tower to the fire?

Answer 1

To find the horizontal distance from the lookout tower to the fire in the valley, we can follow these steps:

1. Determine the vertical distance between the ranger and the fire.

   • The ranger is at an elevation of 4500 ft and the fire is at an elevation of 2000 ft.
   • The vertical distance (\(h\)) is given by: \[ h = 4500 , \text{ft} - 2000 , \text{ft} = 2500 , \text{ft} \]

2. Understanding the angle of depression.

   • The angle of depression is given as 26%. It is common to interpret percentages in this context as angles—however, this percentage seems intended to imply that we use an angle of 26 degrees instead.

3. Using the tangent function to find the horizontal distance.

   • In a right triangle formed by the lookout tower, the horizontal ground distance to the fire, and the vertical distance to the fire, we can use the tangent of the angle of depression (or angle of elevation) to find the horizontal distance.
   • The relationship is given by: \[ \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} \]
   • Here, the opposite side is the vertical distance (\(h\)), and the adjacent side is the horizontal distance (\(d\)).
   • Therefore, we can write: \[ \tan(26^\circ) = \frac{2500 , \text{ft}}{d} \]

4. Rearranging to solve for \(d\): \[ d = \frac{2500 , \text{ft}}{\tan(26^\circ)} \]

5. Calculating \(d\): Using a calculator, we find: \[ \tan(26^\circ) \approx 0.4877 \] Plugging this value in: \[ d = \frac{2500 , \text{ft}}{0.4877} \approx 5125.72 , \text{ft} \]

6. Conclusion: The horizontal distance from the tower to the fire is approximately 5126 feet.