Solve each problem. See Examples 1 and 2.Distance Traveled by a Ship A ship travels 55 km on a bearing of 27° and then travels on a bearing of 117° for 140 km. Find the distance from the starting point to the ending point.

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Step 1: Understand the problem setup. The ship travels in two legs, first 55 km at a bearing of 27° and then 140 km at a bearing of 117°. We need to find the straight-line distance from the starting point to the ending point.

Step 2: Convert the bearings into angles relative to the north-south line. A bearing of 27° is 27° clockwise from north, and a bearing of 117° is 117° clockwise from north.

Step 3: Use the Law of Cosines to find the distance between the starting and ending points. The Law of Cosines is given by: \( c^2 = a^2 + b^2 - 2ab \cdot \cos(C) \), where \( C \) is the angle between the two paths.

Step 4: Calculate the angle between the two paths. Since the bearings are measured clockwise from north, the angle between the two paths is \( 117° - 27° = 90° \).

Step 5: Substitute the known values into the Law of Cosines formula: \( c^2 = 55^2 + 140^2 - 2 \cdot 55 \cdot 140 \cdot \cos(90°) \). Since \( \cos(90°) = 0 \), the formula simplifies to \( c^2 = 55^2 + 140^2 \).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Bearings

Bearings are a way of describing direction using angles measured clockwise from the north. In this problem, the ship's path is defined by two bearings: 27° and 117°. Understanding how to interpret these angles is crucial for visualizing the ship's movement on a coordinate plane and for calculating the resultant distance.

Law of Cosines

The Law of Cosines is a formula used to find the lengths of sides or angles in a triangle when two sides and the included angle are known. In this scenario, after determining the angle between the two paths traveled by the ship, the Law of Cosines can be applied to find the distance from the starting point to the ending point, which is the third side of the triangle formed.

Vector Addition

Vector addition involves combining two or more vectors to determine a resultant vector. In this problem, the ship's journey can be represented as two vectors based on the distances traveled and their respective bearings. By breaking these vectors into their components and adding them, one can find the overall displacement from the starting point to the ending point.

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