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# 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.

Set sail into trigonometric territory with two ships and a towering lighthouse! Unravel the mystery of their distance apart, as observed from angles of 30° and 45°. Explore the seas of mathematics in this captivating problem.

by Maivizhi A

**Updated **Mar 18, 2024

## 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.

## 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 - FAQs

1. What is the angle of elevation?

The angle of elevation is the angle formed between a horizontal line and the line of sight when looking upward at an object. It's commonly used in trigonometry and geometry for calculating heights, distances, and angles in triangles.

2. How is the angle of elevation measured?

The angle of elevation is measured from the horizontal line to the line of sight looking upward.

3. What real-world applications use the angle of elevation?

Real-world applications include surveying, navigation, and physics, where it's used to calculate distances, heights, or lengths based on known angles and measurements.

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