1.9. Example 1.27: Height Calculation of a Lighthouse

From the top of a lighthouse, the angles of depression of two ships are 30° and 45°. The two ships, as it was observed from the top of the lighthouse, were 100 m apart. Find the height of the lighthouse.

In this example, we need to determine the height of a lighthouse based on the angles of depression to two ships located at different angles. The angle of depression is the angle formed by a horizontal line from the observer's eye to the object being observed below the horizon. The two angles given are 30° and 45°, indicating how steeply the line of sight drops down from the top of the lighthouse to the ships. Furthermore, we know the distance between the ships is 100 meters.

Imagine you're standing on the edge of a cliff looking down at two boats in the sea. The first boat is at a 30° angle away from you and the second at a steeper 45°. The cliff's height represents your view point, and you want to accurately calculate how tall the cliff is using the angles and distance to the boats.

Height = Distance / [cot (original angle) – cot (final angle)] Height of the lighthouse = 100 / (cot 30° – cot 45°) = 50 m

Here, we use a specific formula to find the height of the lighthouse. The formula involves the cotangent (cot) of the given angles. The cotangent of an angle in a right triangle is the ratio of the length of the adjacent side to the length of the opposite side. By substituting the angles into the equation and the known distance of the two ships (100 m), we can calculate the height. The resultant calculation shows that the lighthouse height is 50 meters.

Think of using a protractor to measure angles from where you stand on the ground (the boats in our analogy). The mathematical operation of using cotangent effectively helps you calculate how far up you need to go to reach a height that corresponds to that angle, giving you a clear idea of the lighthouse's height above sea level.

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