Problem Statement
A lamp on the top of a light house is visible just above the horizon at a certain station at the sea level. The distance of the top of the light house from the station of observation is 50 km. Find the height of the lamp above sea level.
Solution
Let A be point of observation and B be the bottom of the light house of height ‘h’. Due to the combined effect of curvature and refraction the top of the light house just appears above the horizon.
Using the formula for the combined effect of curvature and refraction:
h = 0.0673 × d²
Where:
- h = height of the lighthouse (in meters)
- d = distance from the observation point (in kilometers)
- 0.0673 = constant factor accounting for Earth’s curvature and atmospheric refraction
Substituting the given value of d = 50 km:
h = 0.0673 × 50²
h = 0.0673 × 2500
h = 168.25 meters
Therefore, the height of the lamp above sea level = 168.25 meters
Detailed Explanation
This problem deals with the concept of horizon distance and how Earth’s curvature affects visibility.
When an observer stands at sea level (point A), their line of sight follows a tangent to the Earth’s surface. Due to Earth’s curvature, objects beyond a certain distance will fall below this line of sight and become invisible. However, tall structures like lighthouses can be seen from greater distances.
The formula h = 0.0673 × d² accounts for two factors:
- Earth’s Curvature: The Earth curves away from a straight line of sight
- Atmospheric Refraction: Light rays bend slightly as they travel through the atmosphere
In this problem, since the lamp is “just visible above the horizon,” it means the line of sight from point A is exactly tangent to the Earth’s surface and reaches the top of the lighthouse at the 50 km distance.
By applying the formula with d = 50 km, we calculate that the lighthouse must be 168.25 meters tall to be visible from that distance, compensating exactly for the Earth’s curvature and atmospheric refraction effects.
