To solve this problem, we need to determine the angle of depression from the bird to the observer. This angle can be calculated using basic trigonometry principles.
Here are the details and steps:
1. Understand the given values:
- Distance from the observer to the bird (`d_observed`): 900 feet (horizontal distance)
- Height of the tower (`h_tower`): 200 feet (vertical distance)
2. Identify the right triangle:
- The triangle formed has the observer at the base (`O`), the top of the tower (`T`) at one vertex, and the bird (`B`) at the other.
- `OT` is the vertical side of the tower (200 feet).
- `OB` is the horizontal distance from the observer to where the bird is spotted (900 feet).
3. Use the tangent function to find the angle of depression (`θ`):
- Tangent of an angle in a right triangle is the ratio of the opposite side to the adjacent side.
- Here, the opposite side is the height of the tower (200 feet), and the adjacent side is the distance from the observer to the bird (900 feet).
4. Set up the equation:
[tex]\[
\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} = \frac{200}{900}
\][/tex]
5. Calculate the angle using the arctangent function:
[tex]\[
\theta = \arctan\left(\frac{200}{900}\right)
\][/tex]
6. Convert the result from radians to degrees:
After performing the calculations:
[tex]\[
\theta \approx 12.528807709151511 \text{ degrees}
\][/tex]
So, the angle of depression from the bird to the observer is approximately [tex]\(12.53^{\circ}\)[/tex].
