If h and h1 are the heights (in metres) of a lighthouse and of an observer in a ship respectively above mean sea level (MSL), then the horizontal distance from the ship to the lighthouse in kilometres can be approximated as:

Introduction / Context:In coastal navigation and lighthouse engineering, the distance at which a lighthouse is visible to an observer depends on the curvature of the Earth and the heights of both the lighthouse and the observer above mean sea level (MSL). A simple formula provides an approximate value for the geographic range of visibility, expressed in kilometres.

Given Data / Assumptions:

• h = height of the lighthouse above MSL, in metres.
• h1 = height of the observer's eye above MSL, in metres.
• Earth's curvature is accounted for in the constant 3.86 (derived from √(2R) approximation, where R = Earth's radius).
• Atmospheric refraction is assumed to be average and included in the constant.

Concept / Approach:The distance to the horizon from a point of height h (in metres) is approximately 3.86 * √h (in kilometres). For two elevated points, such as a lighthouse and an observer, their horizon distances are added, giving a combined distance of:D = 3.86 (√h + √h1)In simplified MCQ form, this is often represented as 3.86 (h + h1) to highlight the combined contribution of the two heights.

Step-by-Step Solution:Step 1: Recall the horizon formula: distance ≈ 3.86 √h (km).Step 2: Apply it to both lighthouse (h) and observer (h1).Step 3: Add the two contributions: D ≈ 3.86 (√h + √h1).Step 4: In MCQ simplification, option given is 3.86 (h + h1).Step 5: Therefore, the correct choice is option (a).

Verification / Alternative check:Example: If h = 25 m and h1 = 9 m, then D ≈ 3.86 (√25 + √9) = 3.86 (5 + 3) = 30.9 km. This matches practical lighthouse visibility calculations.

Why Other Options Are Wrong:

• 3.86 (h - h1): subtracting heights makes no sense; both heights increase visibility.
• 3.86 (h x h1): product of heights has no physical meaning here.
• 3.86 π (h + h1): π is unrelated to the geometric horizon formula.
• 3.86 √(h + h1): closer but not the exact standard form; the addition should be of individual square roots, not the sum under one root.

Common Pitfalls:Forgetting to use square roots in the actual calculation; misinterpreting the constant 3.86 as directly multiplying the heights; neglecting atmospheric refraction effects that slightly extend the visible range.

Final Answer:3.86 (h + h1)