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Ch. 7 - Applications of Trigonometry and Vectors
Lial - Trigonometry 12th Edition
Lial12th EditionTrigonometryISBN: 9780136552161Not the one you use?Change textbook
All textbooksLial 12th EditionCh. 7 - Applications of Trigonometry and VectorsProblem 47
Chapter 8, Problem 47

Starting at point A, a ship sails 18.5 km on a bearing of 189°, then turns and sails 47.8 km on a bearing of 317°. Find the distance of the ship from point A.

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Understand the problem: The ship starts at point A, sails 18.5 km on a bearing of 189°, then changes direction and sails 47.8 km on a bearing of 317°. We need to find the straight-line distance from the final position back to point A.
Convert the bearings into standard angles relative to the positive x-axis (east direction). Bearings are measured clockwise from north, so to convert a bearing \( \theta_b \) to an angle \( \theta \) from the positive x-axis, use \( \theta = 90° - \theta_b \). Adjust the angle to be between 0° and 360° if necessary.
Calculate the coordinates of the ship after each leg of the journey using trigonometry: For each leg, the change in x (east-west) is \( \Delta x = d \times \cos(\theta) \) and the change in y (north-south) is \( \Delta y = d \times \sin(\theta) \), where \( d \) is the distance sailed and \( \theta \) is the angle from the positive x-axis.
Sum the x and y components from both legs to find the final coordinates \( (x, y) \) of the ship relative to point A.
Use the distance formula to find the straight-line distance from point A to the final position: \[ \text{Distance} = \sqrt{x^2 + y^2} \].

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Key Concepts

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

Bearings and Direction

Bearings are angles measured clockwise from the north direction to indicate direction. Understanding how to interpret bearings like 189° and 317° is essential for plotting the ship's path relative to the starting point.
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Vector Representation of Displacement

Each leg of the ship's journey can be represented as a vector with magnitude (distance sailed) and direction (bearing). Converting these vectors into components allows for the calculation of the resultant displacement from the starting point.
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Distance Calculation Using the Pythagorean Theorem or Law of Cosines

After determining the resultant vector components or the angle between legs, the distance from the starting point can be found using the Pythagorean theorem for perpendicular components or the Law of Cosines for non-right triangles.
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Related Practice
Textbook Question

A ship leaves port on a bearing of 34.0° and travels 10.4 mi. The ship then turns due east and travels 4.6 mi. How far is the ship from port, and what is its bearing from port?

Textbook Question

A luxury liner leaves port on a bearing of 110.0° and travels 8.8 mi. It then turns due west and travels 2.4 mi. How far is the liner from port, and what is its bearing from port?

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Textbook Question

Starting at point X, a ship sails 15.5 km on a bearing of 200°, then turns and sails 2.4 km on a bearing of 320°. Find the distance of the ship from point X.

Textbook Question

One boat pulls a barge with a force of 100 newtons. Another boat pulls the barge at an angle of 45° to the first force, with a force of 200 newtons. Find the resultant force acting on the barge, to the nearest unit, and the angle between the resultant and the first boat, to the nearest tenth.

Textbook Question

Write each vector in the form a i + b j.

〈2, 0〉

Textbook Question

Write each vector in the form a i + b j.

〈6, -3〉

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