Geodetic visibility of a lighthouse over the sea horizon A lighthouse has a height of 120 m above mean sea level and is just visible above the horizon from a ship at sea level. Considering the combined correction for Earth’s curvature and standard atmospheric refraction, determine the correct horizontal distance between the ship and the lighthouse, expressed in metres.

Introduction:Surveying and geodesy problems often involve visibility to the horizon. When an object such as a lighthouse is just visible, the line of sight grazes the Earth’s curved surface. The practical formula for horizon distance incorporates a standard refraction allowance, which slightly bends light downward and increases the visible range compared to a vacuum condition.

Given Data / Assumptions:

• Lighthouse height h = 120 m (observer at near sea level).
• Standard combined correction for curvature and refraction is used.
• Distance sought is along the surface (plan distance) in metres.

Concept / Approach:

With standard refraction, the horizon distance from a height h (in metres) is commonly taken as d(km) = 3.86 * sqrt(h). Without refraction, a typical factor is about 3.57. Here we use 3.86 as the combined curvature–refraction factor to compute the visibility distance to the horizon for h = 120 m and then convert to metres.

Step-by-Step Solution:

Verification / Alternative check:

Using the no-refraction factor (3.57) would give about 39.1 km (≈ 39098 m), which matches one distractor and confirms why including refraction increases the distance to about 42.2 km.

Why Other Options Are Wrong:

39.098 and 39098 correspond to the no-refraction estimate; 42.226 (metres) misplaces units by three orders of magnitude; 40000 is a rough round-number guess, not the refined value.

Common Pitfalls:

Confusing kilometres with metres; using the curvature-only factor (3.57) instead of the combined curvature–refraction factor; forgetting that visibility distance scales with the square root of height.

Final Answer:

42226