Final answer:
The angle of a person's eyes to the top of a 200-foot building, standing at the end of an 80-foot shadow with eyes 4 feet above ground level, is approximately 67.98 degrees.
Step-by-step explanation:
To find the angle of a person's eyes to the top of a 200-foot building when standing at the end of an 80-foot shadow, with their eye height being 4 feet above the ground, we can use trigonometric functions, specifically the tangent function, which relates the angle θ to the opposite side over the adjacent side in a right-angled triangle. The building and its shadow, along with the viewer's eye line, form a right-angled triangle.
First, we adjust the building's height to account for the viewer's eye level:
- Building's actual height = 200 feet
- Viewer's eye level = 4 feet
- Adjusted building height = Building's actual height - Viewer's eye level = 196 feet
Now, we apply the tangent function:
- θ = tan⁻¹(opposite/adjacent)
- θ = tan⁻¹(196 feet / 80 feet)
- θ = tan⁻¹(2.45)
Using a calculator, we find that θ is approximately:
Therefore, the angle of the person's eyes to the top of the building is approximately 67.98 degrees to the nearest hundredth of a degree.