An Airplane Flies 200 Km Due West From City A To City B And Then 240 Km In The Direction Of 29.5° North Of West From City B To City C. (a) In Straight-line Distance, How Far Is City C From City A? (b) Relative To City A, In What Direction Is City C?

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Introduction

In this article, we will explore the concept of distance and direction in physics, using a real-world scenario as an example. We will calculate the straight-line distance between two cities, A and C, and determine the direction of city C relative to city A.

The Journey Begins

Let's break down the airplane's journey into two segments:

  • The first segment is a 200 km flight due west from city A to city B.
  • The second segment is a 240 km flight in the direction of 29.5° north of west from city B to city C.

Calculating the Straight-Line Distance

To calculate the straight-line distance between city A and city C, we can use the law of cosines. The law of cosines states that for any triangle with sides of length a, b, and c, and angle C opposite side c, the following equation holds:

c² = a² + b² - 2ab * cos(C)

In this case, we have:

  • a = 200 km (distance from A to B)
  • b = 240 km (distance from B to C)
  • C = 29.5° (angle between sides a and b)

Plugging these values into the equation, we get:

c² = 200² + 240² - 2 * 200 * 240 * cos(29.5°) c² = 40000 + 57600 - 96000 * cos(29.5°) c² = 97600 - 96000 * cos(29.5°)

Using a calculator to evaluate the cosine function, we get:

c² = 97600 - 96000 * 0.874 c² = 97600 - 83840 c² = 13760

Taking the square root of both sides, we get:

c = √13760 c ≈ 371.5 km

Calculating the Direction

To determine the direction of city C relative to city A, we need to find the angle between the line connecting A and C and the x-axis (east-west direction).

We can use the law of sines to find this angle:

sin(A) = (a / c) * sin(C) sin(A) = (200 / 371.5) * sin(29.5°) sin(A) ≈ 0.539

Using a calculator to find the inverse sine, we get:

A ≈ arcsin(0.539) A ≈ 32.5°

This means that city C is located 32.5° north of west from city A.

Conclusion

In this article, we calculated the straight-line distance between city A and city C, and determined the direction of city C relative to city A. We used the law of cosines to find the distance and the law of sines to find the direction.

The straight-line distance between city A and city C is approximately 371.5 km, and city C is located 32.5° north of west from city A.

Discussion

This problem is a classic example of a right triangle problem in physics. The law of cosines and the law of sines are essential tools for solving problems involving right triangles.

In real-world applications, these laws are used in a variety of fields, including navigation, engineering, and physics. For example, in navigation, the law of cosines is used to calculate distances and directions between two points on the Earth's surface.

In engineering, the law of cosines is used to design and optimize systems, such as bridges and buildings. In physics, the law of cosines is used to study the behavior of particles and systems in various fields, such as mechanics and electromagnetism.

References

  • "Law of Cosines" by Math Open Reference
  • "Law of Sines" by Math Open Reference
  • "Right Triangle Problems" by Khan Academy

Further Reading

  • "Trigonometry" by MIT OpenCourseWare
  • "Physics for Scientists and Engineers" by Paul A. Tipler
  • "Calculus" by Michael Spivak

Note: The references and further reading section are not included in the word count.

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Introduction

In our previous article, we explored the concept of distance and direction in physics, using a real-world scenario as an example. We calculated the straight-line distance between two cities, A and C, and determined the direction of city C relative to city A.

In this article, we will answer some frequently asked questions related to the airplane's journey and right triangle problems.

Q&A

Q: What is the law of cosines, and how is it used in right triangle problems?

A: The law of cosines is a mathematical formula that relates the lengths of the sides of a triangle to the cosine of one of its angles. It is used to calculate the length of the third side of a triangle when the lengths of the other two sides and the angle between them are known.

Q: How is the law of sines used in right triangle problems?

A: The law of sines is a mathematical formula that relates the lengths of the sides of a triangle to the sines of its angles. It is used to calculate the length of a side of a triangle when the lengths of the other two sides and the angle between them are known.

Q: What is the difference between the law of cosines and the law of sines?

A: The law of cosines is used to calculate the length of the third side of a triangle when the lengths of the other two sides and the angle between them are known. The law of sines is used to calculate the length of a side of a triangle when the lengths of the other two sides and the angle between them are known.

Q: How is the airplane's journey related to right triangle problems?

A: The airplane's journey is a classic example of a right triangle problem. The law of cosines and the law of sines are used to calculate the distance and direction of the airplane's journey.

Q: What are some real-world applications of right triangle problems?

A: Right triangle problems have many real-world applications, including navigation, engineering, and physics. For example, in navigation, the law of cosines is used to calculate distances and directions between two points on the Earth's surface.

Q: How can I use the law of cosines and the law of sines in my daily life?

A: You can use the law of cosines and the law of sines in your daily life by applying them to real-world problems. For example, if you are planning a trip and need to calculate the distance and direction of your journey, you can use the law of cosines and the law of sines.

Conclusion

In this article, we answered some frequently asked questions related to the airplane's journey and right triangle problems. We hope that this article has provided you with a better understanding of the law of cosines and the law of sines, and how they are used in right triangle problems.

Discussion

Right triangle problems are an essential part of mathematics and physics. The law of cosines and the law of sines are fundamental tools for solving problems involving right triangles.

In real-world applications, right triangle problems are used in a variety of fields, including navigation, engineering, and physics. For example, in navigation, the law of cosines is used to calculate distances and directions between two points on the Earth's surface.

References

  • "Law of Cosines" by Math Open Reference
  • "Law of Sines" by Math Open Reference
  • "Right Triangle Problems" by Khan Academy

Further Reading

  • "Trigonometry" by MIT OpenCourseWare
  • "Physics for Scientists and Engineers" by Paul A. Tipler
  • "Calculus" by Michael Spivak

Note: The references and further reading section are not included in the word count.