To solve for the angle of depression from the bird (B) to the observer (O), we can use trigonometric principles. Given the distances are as follows:
- Distance from the bird to the tree (B to T) is 5000 feet.
- Distance from the bird to the observer (B to O) is 13000 feet.
We need to determine the angle of depression from the bird to the observer. This angle is formed between the line connecting the bird to the observer and the horizontal line drawn through the bird. This can be understood as the angle such that its cosine is the ratio of the adjacent side (distance from the bird to the tree) to the hypotenuse (distance from the bird to the observer).
Here is the step-by-step process to solve this:
1. Identify known values:
- Adjacent side (distance from the bird to the tree): [tex]\( \text{adjacent} = 5000 \)[/tex] feet
- Hypotenuse (distance from the bird to the observer): [tex]\( \text{hypotenuse} = 13000 \)[/tex] feet
2. Calculate the cosine of the angle:
[tex]\[
\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}}
\][/tex]
[tex]\[
\cos(\theta) = \frac{5000}{13000}
\][/tex]
3. Solve for the angle [tex]\(\theta\)[/tex]:
[tex]\[
\theta = \cos^{-1}\left(\frac{5000}{13000}\right)
\][/tex]
Using the values provided:
[tex]\[
\theta = \cos^{-1}\left(\frac{5000}{13000}\right)
\][/tex]
4. Convert the angle from radians to degrees:
[tex]\[
\theta = \theta \times \left(\frac{180}{\pi}\right)
\][/tex]
5. Solution:
The value of the angle [tex]\(\theta\)[/tex] is approximately:
[tex]\[
67.38^\circ
\][/tex]
The angle of depression from the bird to the observer is approximately [tex]\( 67.38^\circ \)[/tex].
Thus, the correct answer is:
[tex]\[ 67.38^\circ \][/tex]
