Two ships are sailing in the sea on the two sides of a lighthouse. The angle of elevation of the top of the lighthouse is observed from the ships are 30° and 45° respectively.

Two ships are sailing in the sea on the two sides of a lighthouse. The angle of elevation of the top of the lighthouse is observed from the ships are 30° and 45° respectively. If the lighthouse is 100 m high, the distance between the two ships is

The distance between the two ships is approximately 273 meters.

To solve this problem, we can use trigonometry, specifically the tangent function.

Let's denote:

• h be the height of the lighthouse (which is given as 100 m).
• d1 be the distance between the first ship and the lighthouse.
• d2 be the distance between the second ship and the lighthouse.

Now, we can set up equations based on the tangent of the angles of elevation:

For the first ship: tan(30°) = h / d1

For the second ship: tan(45°) = h / d2

We know h = 100 m. Let's solve these equations for d1 and d2:

For the first ship: d1 = h / tan(30°) = 100 / tan(30°)

For the second ship: d2 = h / tan(45°) = 100 / tan(45°)

Now, we can calculate these values:

d1 = 100 / tan(30°) ≈ 100 / 0.577 ≈ 173.21 m

d2 = 100 / tan(45°) ≈ 100 / 1 = 100 m

Finally, the distance between the two ships is the sum of these distances:

Distance between the ships = d1 + d2 ≈ 173.21 m + 100 m = 273.21 m

So, the distance between the two ships is approximately 273.21 meters.

What is Angle of Elevation?

In mathematics, the angle of elevation refers to the angle formed between a horizontal line and the line of sight when an observer looks upward at an object. It is typically used in trigonometry and geometry, especially in problems involving heights, distances, and angles in triangles.

For instance, if you're standing on level ground and looking up at the top of a tree, the angle formed between the ground (horizontal line) and your line of sight to the top of the tree is the angle of elevation.

The concept is often used in real-world applications such as surveying, navigation, and physics to calculate distances, heights, or lengths indirectly based on known angles and measurements.