Navigating Trigonometry with Real-World Applications: Solving for Distance Between Boats

In a coastal scenario, a lighthouse standing at 200 meters tall helps sailors navigate the seas. By understanding the principles of right triangles and trigonometry, we can solve for the distance between two boats positioned on either side of the lighthouse. This article explores the application of these mathematical concepts in a practical, real-world setting, providing a step-by-step guide to determining the distance between the two boats.

Understanding the Scenario

Imagine standing atop a lighthouse, 200 meters tall, with an angle of depression of 45 degrees to two boats in the sea, one on either side of the lighthouse. The angle of depression is the angle formed between the horizontal line from the observer's eye and the line of sight to a point below. In this case, the angle of depression is 45 degrees, which is the same as the angle of elevation from the boats to the top of the lighthouse.

Identifying the Triangle

To solve for the distance between the two boats, we need to identify and analyze the right triangles formed by the lighthouse, the boats, and the angle of depression.

The height of the lighthouse is 200 meters. The angle of depression to each boat is 45 degrees.

Using Trigonometry

The angle of depression to each boat corresponds to an angle of elevation from each boat to the top of the lighthouse, which is also 45 degrees. In a right triangle with a 45-degree angle, the opposite side (height of the lighthouse) is equal to the adjacent side (distance from the base of the lighthouse to each boat).

Calculating the Distance to Each Boat

Let's denote the distance from the base of the lighthouse to one boat as d. For a 45-degree angle in a right triangle, the opposite side (height of the lighthouse) is equal to the adjacent side (distance from the base of the lighthouse to each boat). Therefore:

[tan 45^circ frac{h}{d} 1 Rightarrow frac{200}{d} 1 Rightarrow d 200 text{ m}]

Calculating the Total Distance Between the Two Boats

Since there are two boats, one on each side of the lighthouse, the total distance D between the two boats is the sum of the distances from the base of the lighthouse to each boat:

[D d d 200 200 400 text{ m}]

Visualizing the Solution

The illustration below helps visualize the scenario. In the figure, O is the lighthouse, and A and B are the boats. OAC and OBC are right-angled isosceles triangles. Since the height of the lighthouse is 200 meters, and the angle of depression (and elevation) is 45 degrees, the distance from the base of the lighthouse to each boat is 200 meters. Therefore, the total distance between the two boats is 400 meters.

Illustration of the right-angled isosceles triangles formed by the lighthouse and the boats.

Alternative Method

There are two alternative ways to approach this problem:

Consider the top of the lighthouse as point A, and the base as point B. If points Y and Z are the positions of the ships, then YBZ forms a straight line (ignoring the curvature of the earth). Triangle ABY and ABZ are similar isosceles triangles because the angles at A and B are 45 degrees. Therefore, YB AB and ZB AB, so YB ZB 200 m, and the boats are 400 meters apart. Using the trigonometric relationship in triangle ABY, we get YB/200 tan 45, and since tan 45 1, YB 200. The same applies for BZ, making the total distance between the two boats 400 meters.

Conclusion

By applying the principles of right triangles and trigonometry, we can accurately determine the distance between two boats using the height of a lighthouse and the angle of depression. This example demonstrates the practical application of mathematical concepts in navigation and maritime contexts.