To determine how far the bird (B) is from its nest (N), we'll consider the given problem using trigonometric principles.
1. Understand the given values:
- The angle of elevation from the observer (O) to the bird (B) is [tex]\(35^{\circ}\)[/tex].
- The distance from the observer (O) to the bird (B) is 21,000 feet.
2. Visualization:
Let's visualize this problem with a right-angled triangle:
- The hypotenuse of the triangle (OB) is the distance from the observer to the bird, which is 21,000 feet.
- The angle at O is [tex]\(35^{\circ}\)[/tex].
- We need to find the horizontal distance (BN) from the bird to the nest.
3. Relevant Trigonometric Function:
To find the horizontal distance, we need to use the cosine function, as cosine relates the adjacent side to the hypotenuse in a right-angled triangle:
[tex]\[
\cos(\text{angle}) = \frac{\text{adjacent}}{\text{hypotenuse}}
\][/tex]
Here, the adjacent side is BN, the angle is [tex]\(35^{\circ}\)[/tex], and the hypotenuse is OB (21,000 feet).
4. Set up the equation:
[tex]\[
\cos(35^{\circ}) = \frac{BN}{21,000}
\][/tex]
5. Solve for BN:
[tex]\[
BN = 21,000 \times \cos(35^{\circ})
\][/tex]
6. Use the cosine value:
Calculate [tex]\( \cos(35^{\circ}) \)[/tex] which is approximately 0.8192.
7. Multiply to find BN:
[tex]\[
BN = 21,000 \times 0.8192 = 17,202 \text{ feet}
\][/tex]
Therefore, the horizontal distance from the bird to the nest is 17,202 feet. The correct answer is:
- 17,202 feet
