Calculating Horizontal Distance

Oct 13, 2025 by miodeya.com 133 views

===========================================================

Introduction

In this article, we will delve into the world of trigonometry and explore how to calculate the horizontal distance of a boat from a lighthouse using the angle of elevation and the height of the beacon. This problem is a classic example of a right-angled triangle problem, where we can use trigonometric ratios to find the unknown side length.

The Problem

A boat is heading towards a lighthouse whose beacon-light is 131 feet above the water. The boat's crew measures the angle of elevation to the beacon to be 7 degrees. We need to find the ship's horizontal distance from the lighthouse.

Understanding the Problem

To solve this problem, we need to understand the concept of right-angled triangles and trigonometric ratios. A right-angled triangle is a triangle with one angle equal to 90 degrees. In this case, the lighthouse is the vertex of the right angle, and the boat is at a distance from the lighthouse. The angle of elevation is the angle between the line of sight from the boat to the beacon and the horizontal plane.

Trigonometric Ratios

Trigonometric ratios are used to relate the angles and side lengths of a right-angled triangle. The three basic trigonometric ratios are:

Sine (sin): The ratio of the length of the side opposite the angle to the length of the hypotenuse (the side opposite the right angle).
Cosine (cos): The ratio of the length of the side adjacent to the angle to the length of the hypotenuse.
Tangent (tan): The ratio of the length of the side opposite the angle to the length of the side adjacent to the angle.

Calculating Horizontal Distance

To calculate the horizontal distance of the boat from the lighthouse, we can use the tangent ratio. The tangent of an angle is equal to the ratio of the length of the side opposite the angle to the length of the side adjacent to the angle. In this case, the angle is 7 degrees, and the length of the side opposite the angle is the height of the beacon (131 feet).

We can set up the following equation:

tan(7°) = opposite side (131 feet) / adjacent side (horizontal distance)

To solve for the horizontal distance, we can rearrange the equation:

horizontal distance = opposite side (131 feet) / tan(7°)

Calculating the Horizontal Distance

Now, we can calculate the horizontal distance using a calculator or a trigonometric table. The tangent of 7 degrees is approximately 0.1235. We can plug this value into the equation:

horizontal distance = 131 feet / 0.1235

horizontal distance ≈ 1060.3 feet

Conclusion

In this article, we have used trigonometric ratios to calculate the horizontal distance of a boat from a lighthouse. We have set up an equation using the tangent ratio and solved for the horizontal distance. The result is approximately 1060.3 feet. This problem is a classic example of a right-angled triangle problem, and it demonstrates the importance of trigonometric ratios in solving real-world problems.

Additional Information

Trigonometric Ratios: Trigonometric ratios are used to relate the angles and side lengths of a right-angled triangle. The three basic trigonometric ratios are sine, cosine, and tangent.
Right-Angled Triangles: A right-angled triangle is a triangle with one angle equal to 90 degrees. In this case, the lighthouse is the vertex of the right angle, and the boat is at a distance from the lighthouse.
Angle of Elevation: The angle of elevation is the angle between the line of sight from the boat to the beacon and the horizontal plane.

References

Trigonometry: Trigonometry is a branch of mathematics that deals with the relationships between the sides and angles of triangles. It is used to solve problems involving right-angled triangles.
Right-Angled Triangles: A right-angled triangle is a triangle with one angle equal to 90 degrees. In this case, the lighthouse is the vertex of the right angle, and the boat is at a distance from the lighthouse.
Trigonometric Ratios: Trigonometric ratios are used to relate the angles and side lengths of a right-angled triangle. The three basic trigonometric ratios are sine, cosine, and tangent.

=====================================

Frequently Asked Questions

In this article, we will answer some of the most frequently asked questions related to the problem of calculating the horizontal distance of a boat from a lighthouse.

Q: What is the angle of elevation in this problem?

A: The angle of elevation in this problem is 7 degrees.

Q: What is the height of the beacon in this problem?

A: The height of the beacon in this problem is 131 feet.

Q: What is the horizontal distance of the boat from the lighthouse?

A: The horizontal distance of the boat from the lighthouse is approximately 1060.3 feet.

Q: How did you calculate the horizontal distance?

A: We used the tangent ratio to calculate the horizontal distance. The tangent of an angle is equal to the ratio of the length of the side opposite the angle to the length of the side adjacent to the angle. In this case, the angle is 7 degrees, and the length of the side opposite the angle is the height of the beacon (131 feet).

Q: What is the formula for calculating the horizontal distance?

A: The formula for calculating the horizontal distance is:

horizontal distance = opposite side (131 feet) / tan(7°)

Q: Can I use this formula to calculate the horizontal distance for any angle?

A: Yes, you can use this formula to calculate the horizontal distance for any angle. However, you will need to know the height of the beacon and the angle of elevation.

Q: What is the significance of the tangent ratio in this problem?

A: The tangent ratio is significant in this problem because it allows us to relate the angle of elevation to the horizontal distance of the boat from the lighthouse.

Q: Can I use other trigonometric ratios to calculate the horizontal distance?

A: Yes, you can use other trigonometric ratios to calculate the horizontal distance. However, the tangent ratio is the most convenient ratio to use in this problem.

Q: What is the relationship between the angle of elevation and the horizontal distance?

A: The angle of elevation and the horizontal distance are related through the tangent ratio. As the angle of elevation increases, the horizontal distance decreases.

Q: Can I use this formula to calculate the horizontal distance for a lighthouse with a different height?

A: Yes, you can use this formula to calculate the horizontal distance for a lighthouse with a different height. However, you will need to know the height of the beacon and the angle of elevation.

Q: What is the importance of trigonometry in this problem?

A: Trigonometry is important in this problem because it allows us to relate the angles and side lengths of a right-angled triangle. In this case, we use the tangent ratio to calculate the horizontal distance of the boat from the lighthouse.

Q: Can I use this formula to calculate the horizontal distance for a boat moving at a different speed?

A: No, you cannot use this formula to calculate the horizontal distance for a boat moving at a different speed. This formula only works for a stationary boat.

Q: What is the relationship between the angle of elevation and the height of the beacon?

A: The angle of elevation and the height of the beacon are related through the tangent ratio. As the angle of elevation increases, the height of the beacon decreases.

Q: Can I use this formula to calculate the horizontal distance for a lighthouse with a different angle of elevation?

A: Yes, you can use this formula to calculate the horizontal distance for a lighthouse with a different angle of elevation. However, you will need to know the height of the beacon and the new angle of elevation.

Q: What is the significance of the right-angled triangle in this problem?

A: The right-angled triangle is significant in this problem because it allows us to use trigonometric ratios to relate the angles and side lengths of the triangle.

Q: Can I use this formula to calculate the horizontal distance for a boat moving in a different direction?

A: No, you cannot use this formula to calculate the horizontal distance for a boat moving in a different direction. This formula only works for a boat moving directly towards the lighthouse.

Q: What is the relationship between the angle of elevation and the horizontal distance?

A: The angle of elevation and the horizontal distance are related through the tangent ratio. As the angle of elevation increases, the horizontal distance decreases.

Q: Can I use this formula to calculate the horizontal distance for a lighthouse with a different shape?

A: No, you cannot use this formula to calculate the horizontal distance for a lighthouse with a different shape. This formula only works for a lighthouse with a circular shape.

Q: What is the importance of trigonometry in this problem?

Q: Can I use this formula to calculate the horizontal distance for a boat moving at a different speed and direction?

A: No, you cannot use this formula to calculate the horizontal distance for a boat moving at a different speed and direction. This formula only works for a stationary boat moving directly towards the lighthouse.