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Chapter 7: Problem 43

A plane flies 500 kilometers with a bearing of \(316^{\circ}\) (clockwise from north) from Naples to Elgin. The plane then flies 720 kilometers from Elgin to Canton (see figure). Canton is due west of Naples. Find the bearing of the flight from Elgin to Canton.

Short Answer

Expert verified
The bearing of the flight from Elgin to Canton is calculated to be approximately \(270^{\circ}\) west.

Step by step solution

01

Formulate the Triangle

Formulate a triangle using the positions of Naples, Elgin, and Canton. Elgin (E) is at a bearing of \(316^{\circ}\) from Naples (N) and Canton (C) is due west of Naples. The bearing of the flight from Elgin to Canton is therefore the interior angle at E in the triangle NEC.
02

Find the Angle at Naples

To find the bearing from Elgin to Canton, first determine the angle at Naples. This can be done by subtracting the bearing from Naples to Elgin from \(360^{\circ}\) (since bearings are measured clockwise from North). This gives \(360^{\circ} - 316^{\circ} = 44^{\circ}\) as the angle at N.
03

Use the Law of Cosine to Find the Angle at Elgin

With the distances from Naples (N) to Elgin (E) and from Elgin to Canton (C) known, as well as the angle at N, we can use the law of cosines to find angle NEC. Thus, \(\cos{NEC} = (NE^2 + EC^2 - NC^2) / 2*NE*EC\). With values given, this amounts to \(\cos{NEC} = \frac{{500^2 + 720^2 - [(\sqrt{{500^2 + 720^2}})]^2}}{{2*500*720}} \). Solving this gives the angle NEC.
04

Find the Bearing from Elgin to Canton

Finally, subtract the angle at NEC from \(360^{\circ}\) to get the bearing from Elgin to Canton because the bearing is measured clockwise from north.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Law of Cosines
The Law of Cosines is a powerful tool in trigonometry that helps us relate the lengths of the sides of a triangle to its angles. It is especially useful when dealing with non-right triangles.
The formula is given by: \[ c^2 = a^2 + b^2 - 2ab \cos(C)\] where \(c\) is the side opposite the angle \(C\), and \(a\) and \(b\) are the other two sides of the triangle.
This allows us to solve for unknown angles or side lengths when some values are known.
In the exercise, the law of cosines helped us find the angle at Elgin (E) by using the known lengths of the sides (Naples to Elgin and Elgin to Canton) and the previously calculated angle at Naples. By plugging these values into the law of cosines, we calculated the cosine of angle NEC. This crucial step enabled us to later determine the bearing of the flight from Elgin to Canton.
Bearing Angles
Understanding bearings is key in navigation, as these angles provide directions based on the cardinal points (North, East, South, and West). Bearings are typically measured in degrees clockwise from North.
For example, a bearing of \(316^{\circ}\) indicates a direction close to Northwest, but slightly more towards the North.
In the exercise, we used bearings to define the direction of flight between cities. From Naples to Elgin, the plane flew on a \(316^{\circ}\) bearing. With this, we determined other directional relationships, specifically finding the angle at Naples. By knowing this initial orientation, it helped build a clearer geometrical understanding of the triangle formed by the flight paths.
Triangle Formation in Trigonometry
In trigonometry, forming triangles is the basis for determining unknown quantities鈥攂e it angles or side lengths. When we have a real-world problem involving directions or distances, like in flight navigation, we conceptualize it as geometric shapes, often triangles.
In this exercise, we considered the cities Naples, Elgin, and Canton as points forming a triangle. Naples is the northernmost point, while Canton is directly west of Naples, establishing a horizontal reference.
The positioning was key: knowing Elgin鈥檚 bearing from Naples at \(316^{\circ}\) allowed us to complete the triangle. With this blueprint, the challenge was to find other angles or sides using trigonometric laws such as the Law of Cosines. Understanding the layout of this triangle helped in visualizing the navigation paths and solving the problem intricately and accurately.

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