The effect of Earth's curvature on height observations is equal to which expression, where K is the horizontal distance in kilometers?

• A. 0.0675K2
• B. 0.0885K2
• C. 0.0785K2
• D. 0.0110K2

Answer: (C)

The height error from Earth's curvature grows with the square of how far you look, because curvature is a second-order geometric effect: as you move farther, the arc length compared to a tangent line introduces a vertical deviation that scales with distance squared.

For a horizontal distance K kilometers, convert to meters and use the Earth's radius R ≈ 6,371,000 meters. The curvature drop (the height difference you’d observe due to the curvature) is about Δh ≈ d^2/(2R). With d = 1000K meters, this becomes Δh ≈ (1,000K)^2 / (2 × 6,371,000) ≈ 1,000,000 K^2 / 12,742,000 ≈ 0.0785 K^2 meters.

So the curvature effect on height observations is approximately 0.0785 × K^2 meters. For example, 10 km gives about 7.85 meters of curvature drop. The other coefficients would imply a different Earth radius and do not match the standard curvature for Earth.