Let's go through the steps to solve the problem:
1. Identify the given information and what is required:
- Distance from the birdwatcher to the first bird: 32 feet.
- Distance from the birdwatcher to the second bird: 45 feet.
- We need to calculate the angle of depression between the second bird and the birdwatcher, rounding the answer to the nearest tenth of a degree.
2. Determine the difference in distances between the two birds:
- The difference in distance between the second bird and the first bird is [tex]\( 45 - 32 = 13 \)[/tex] feet.
3. Visualize the problem:
- Imagine a right-angled triangle where:
- The base represents the distance from the birdwatcher to the point on the ground directly below the first bird (32 feet).
- The vertical difference in distance between the two bird positions represents the height difference (13 feet).
4. Use trigonometry to find the angle of depression:
- We need to use the tangent (tan) of the angle in trigonometry because we have the lengths of the opposite side (difference in distances between the two birds) and the adjacent side (distance to the first bird).
- The formula for the tangent of an angle in a right-angled triangle is:
[tex]\[
\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}
\][/tex]
- Here, θ is the angle of depression we want to find, the opposite side is 13 feet, and the adjacent side is 32 feet:
[tex]\[
\tan(\theta) = \frac{13}{32}
\][/tex]
5. Calculate the angle θ:
- We use the arctangent (inverse tangent) function to find θ:
[tex]\[
\theta = \arctan\left(\frac{13}{32}\right)
\][/tex]
- This calculation gives us the angle in radians. Converting this angle from radians to degrees:
[tex]\[
\theta \approx 22.109448343751673 \text{ degrees}
\][/tex]
6. Round the angle to the nearest tenth:
- Thus, we round 22.109448343751673 to the nearest tenth, which gives us 22.1 degrees.
7. Conclusion:
- Therefore, the angle measure, or angle of depression, between the second bird and the birdwatcher, rounded to the nearest tenth, is [tex]\(22.1^{\circ}\)[/tex].
None of the provided options match this angle, suggesting there might be a miscommunication or error in the problem or the provided options. In any case, the calculated angle of depression is 22.1 degrees.
