A Ship Leaves Port A And Travels 215 Km On A Bearing Of 120 Decrees To Port B And Then Travels On A Bearing Of 165 Decrees Of To Another Port C, 305 Km Away. Calculate The Direction From A To C To The Nearest Km And Find Also The Bearing From Port C To
Introduction
In this article, we will explore the problem of a ship traveling from Port A to Port C, with a stopover at Port B. We will use trigonometry and vector calculations to determine the direction from Port A to Port C and the bearing from Port C to Port A. This problem is a classic example of a two-dimensional vector problem, where we need to calculate the resultant vector and its direction.
The Problem
A ship leaves Port A and travels 215 km on a bearing of 120 degrees to Port B. Then, it travels on a bearing of 165 degrees to another Port C, which is 305 km away. We need to calculate the direction from Port A to Port C and the bearing from Port C to Port A.
Step 1: Convert the Bearings to Standard Position
To solve this problem, we need to convert the bearings to standard position, which is measured counterclockwise from the positive x-axis. We can do this by subtracting the bearing from 360 degrees.
- Bearing from Port A to Port B: 120 degrees
- Bearing from Port B to Port C: 165 degrees
Step 2: Calculate the Vectors
We can represent the vectors as follows:
- Vector AB: 215 km at 120 degrees
- Vector BC: 305 km at 165 degrees
We can calculate the x and y components of each vector using the following formulas:
- x-component = distance * cos(angle)
- y-component = distance * sin(angle)
For Vector AB:
- x-component = 215 km * cos(120 degrees) = -173.2 km
- y-component = 215 km * sin(120 degrees) = 173.2 km
For Vector BC:
- x-component = 305 km * cos(165 degrees) = -143.1 km
- y-component = 305 km * sin(165 degrees) = 266.5 km
Step 3: Calculate the Resultant Vector
To calculate the resultant vector, we need to add the x and y components of the two vectors.
- x-component = -173.2 km + (-143.1 km) = -316.3 km
- y-component = 173.2 km + 266.5 km = 439.7 km
Step 4: Calculate the Direction from Port A to Port C
To calculate the direction from Port A to Port C, we need to calculate the angle of the resultant vector. We can do this using the following formula:
- angle = arctan(y-component / x-component)
Plugging in the values, we get:
- angle = arctan(439.7 km / -316.3 km) = -68.4 degrees
Since the angle is negative, we need to add 360 degrees to get the correct angle.
- angle = -68.4 degrees + 360 degrees = 291.6 degrees
Step 5: Calculate the Bearing from Port C to Port A
To calculate the bearing from Port C to Port A, we need to subtract the angle of the resultant vector from 360 degrees.
- bearing = 360 degrees - 291.6 degrees = 68.4 degrees
Conclusion
In this article, we calculated the direction from Port A to Port C and the bearing from Port C to Port A. We used trigonometry and vector calculations to determine the resultant vector and its direction. The direction from Port A to Port C is 291.6 degrees, and the bearing from Port C to Port A is 68.4 degrees.
Calculations
| Vector | Distance (km) | Angle (degrees) | x-component (km) | y-component (km) |
|---|---|---|---|---|
| AB | 215 | 120 | -173.2 | 173.2 |
| BC | 305 | 165 | -143.1 | 266.5 |
| Resultant | -316.3 | 439.7 |
References
- [1] "Trigonometry" by Michael Corral
- [2] "Vector Calculus" by Michael Corral
Note: The calculations and formulas used in this article are based on the assumption that the bearings are measured counterclockwise from the positive x-axis. If the bearings are measured clockwise, the calculations will be different.
Introduction
In our previous article, we explored the problem of a ship traveling from Port A to Port C, with a stopover at Port B. We used trigonometry and vector calculations to determine the direction from Port A to Port C and the bearing from Port C to Port A. In this article, we will answer some frequently asked questions related to this problem.
Q: What is the bearing from Port A to Port B?
A: The bearing from Port A to Port B is 120 degrees.
Q: What is the distance from Port A to Port B?
A: The distance from Port A to Port B is 215 km.
Q: What is the bearing from Port B to Port C?
A: The bearing from Port B to Port C is 165 degrees.
Q: What is the distance from Port B to Port C?
A: The distance from Port B to Port C is 305 km.
Q: What is the direction from Port A to Port C?
A: The direction from Port A to Port C is 291.6 degrees.
Q: What is the bearing from Port C to Port A?
A: The bearing from Port C to Port A is 68.4 degrees.
Q: How did you calculate the resultant vector?
A: We calculated the resultant vector by adding the x and y components of the two vectors. The x-component of the resultant vector is the sum of the x-components of the two vectors, and the y-component of the resultant vector is the sum of the y-components of the two vectors.
Q: How did you calculate the angle of the resultant vector?
A: We calculated the angle of the resultant vector using the arctan function. The angle is the arctan of the y-component divided by the x-component.
Q: Why did you add 360 degrees to the angle of the resultant vector?
A: We added 360 degrees to the angle of the resultant vector because the angle is negative. In trigonometry, angles are measured counterclockwise from the positive x-axis, and negative angles are measured clockwise from the positive x-axis. By adding 360 degrees, we converted the negative angle to a positive angle.
Q: What is the significance of the bearing from Port C to Port A?
A: The bearing from Port C to Port A is the angle between the line connecting Port C and Port A and the positive x-axis. It is an important piece of information for navigation and orientation.
Q: How can I apply this problem to real-life situations?
A: This problem can be applied to real-life situations such as navigation, orientation, and route planning. For example, if you are a sailor or a pilot, you need to know the direction and bearing from one location to another in order to navigate safely and efficiently.
Conclusion
In this article, we answered some frequently asked questions related to the problem of a ship traveling from Port A to Port C, with a stopover at Port B. We used trigonometry and vector calculations to determine the direction from Port A to Port C and the bearing from Port C to Port A. We hope that this article has provided you with a better understanding of this problem and its applications.
Calculations
| Vector | Distance (km) | Angle (degrees) | x-component (km) | y-component (km) |
|---|---|---|---|---|
| AB | 215 | 120 | -173.2 | 173.2 |
| BC | 305 | 165 | -143.1 | 266.5 |
| Resultant | -316.3 | 439.7 |
References
- [1] "Trigonometry" by Michael Corral
- [2] "Vector Calculus" by Michael Corral
Note: The calculations and formulas used in this article are based on the assumption that the bearings are measured counterclockwise from the positive x-axis. If the bearings are measured clockwise, the calculations will be different.