To solve the problem of finding how far the bird (B) is from its nest (N) given the angle and distance from the observer (O) to the bird, we need to use trigonometric principles.
Here's a step-by-step solution:
1. Understand the problem:
- Observer (O) sees the bird (B) at an angle of [tex]\(55^\circ\)[/tex] from the horizontal line to its nest (N).
- The distance from the observer (O) to the bird (B) is 15,000 feet.
2. Identify the formula:
- In a right triangle, the sine of an angle is given by the ratio of the length of the opposite side to the hypotenuse. Here, we have
[tex]\[
\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}
\][/tex]
- Here, [tex]\(\theta = 55^\circ\)[/tex], the opposite side is the distance from the bird to its nest (N), and the hypotenuse is the distance from the observer (O) to the bird (B).
3. Convert the angle from degrees to radians:
- We convert [tex]\(55^\circ\)[/tex] to radians because trigonometric calculations are typically done in radians. Using the conversion factor [tex]\(\pi \text{ radians} = 180^\circ\)[/tex]:
[tex]\[
\theta = \frac{55 \times \pi}{180} \approx 0.959931 \text{ radians}
\][/tex]
4. Calculate the distance from the bird to its nest:
- Using the sine function:
[tex]\[
\sin(55^\circ) = \sin(0.959931) \approx 0.819152
\][/tex]
- Thus,
[tex]\[
\text{opposite} = \text{hypotenuse} \times \sin(\theta) = 15000 \times 0.819152 \approx 12287.281 \text{ feet}
\][/tex]
5. Round to the nearest whole number:
- The rounded distance is 12,287 feet.
So, the distance from the bird (B) to its nest (N) is approximately 12,287 feet.
Therefore, the correct answer is 12,287 feet.
