Answer
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Step 1:I'll solve this problem step by step, carefully following the LaTeX formatting guidelines:
Step 2:: Understand the Problem
The question asks about the distance to the horizon, which depends on the curvature of the Earth and the observer's height. We'll use the geometric relationship between the Earth's radius and the line of sight.
Step 3:: Identify the Key Formula
- $$h$$ = height of the observer above the Earth's surface
The formula for horizon distance is derived from the right triangle formed by the observer's eye, the Earth's center, and the horizon point: Where:
Step 4:: Determine Typical Observer Height
Let's calculate for an average person standing $$1.7$$ meters ($$0.0017$$ km) tall.
Step 5:: Calculate Horizon Distance
\begin{aligned}
d &= \sqrt{2 \cdot 6,371 \cdot 0.0017 + (0.0017)^{2}} \ &= \sqrt{2 \cdot 6,371 \cdot 0.0017 + 0.00000289} \ &= \sqrt{21.6614 + 0.00000289} \ &= \sqrt{21.66140289} \ &\approx 4.66 \text{ km} \end{aligned}
Step 6:: Verify Calculation
This means a person of average height can see approximately $$4.66$$ kilometers to the horizon before the Earth's curvature blocks the view.
Final Answer
Note: The actual distance can vary slightly based on precise observer height, atmospheric conditions, and local terrain.
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