We need to find the angle of depression from a bird (B) to an observer (O), given the following information:
1. The distance from the tree (T) to the point directly below the bird (B), which is 5,000 feet.
2. The distance from the observer (O) to the bird (B), which is 13,000 feet.
Let's denote:
- The distance from the tree (T) to the point directly below the bird (B) as [tex]\( d_{\text{tree-bird}} = 5000 \)[/tex] feet.
- The distance from the observer (O) to the bird (B) as [tex]\( d_{\text{observer-bird}} = 13000 \)[/tex] feet.
The angle of depression is the angle formed by the line from the bird to the observer and the horizontal line directly below the bird to the point of the observer.
To find the angle of depression, we can use trigonometric functions. Specifically, we can utilize the cosine function since we have the lengths of the adjacent side (the distance between the tree and the point directly below the bird) and the hypotenuse (the distance between the observer and the bird):
[tex]\[ \cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}} \][/tex]
In this case, the adjacent side is [tex]\( d_{\text{tree-bird}} = 5000 \)[/tex] feet, and the hypotenuse is [tex]\( d_{\text{observer-bird}} = 13000 \)[/tex] feet. Plugging in these values:
[tex]\[ \cos(\theta) = \frac{5000}{13000} \][/tex]
Next, we calculate the arccosine (inverse cosine) of this ratio to find the angle [tex]\( \theta \)[/tex]:
[tex]\[ \theta = \cos^{-1}\left(\frac{5000}{13000}\right) \][/tex]
Performing the calculation, we find that:
[tex]\[ \theta \approx 67.38^\circ \][/tex]
Therefore, the angle of depression from the bird to the observer is approximately [tex]\( 67.38^\circ \)[/tex].
Among the provided options, the closest value is:
[tex]\[ \boxed{67.38^\circ} \][/tex]
