A ship leaves port with a bearing of S 40 W. After traveling 7 miles, the ship turns 90 degrees on a bearing of N 50 W for 11 miles. At that time, what is the bearing of the ship from port?
To find the final bearing of the ship from the port, we can use basic trigonometry and geometrical concepts.
Step 1: Determine the relative position of the ship after the first leg of the journey.
- The ship starts with a bearing of S 40 W.
- Convert this bearing to a compass bearing: 180 - 40 = 140 degrees. The bearing is now 140°.
Step 2: Calculate the coordinates after the first leg of the journey.
- The ship has traveled 7 miles with a bearing of 140°.
- We can calculate the change in the north-south (y) coordinate and east-west (x) coordinate using trigonometry.
- The change in the y-coordinate is given by: y = 7 * sin(140°).
- The change in the x-coordinate is given by: x = 7 * cos(140°).
- Calculate these values using a scientific calculator or trigonometry tables.
Step 3: Determine the position after the second leg of the journey.
- The ship turns 90 degrees on a bearing of N 50 W for 11 miles.
- Subtracting 50 from 360 gives us the final bearing: 360 - 50 = 310°.
- This bearing is now measured clockwise from the north.
Step 4: Calculate the coordinates after the second leg of the journey.
- The ship has traveled 11 miles with a bearing of 310°.
- Similarly, calculate the change in the y and x coordinates using:
y = 11 * sin(310°) and x = 11 * cos(310°).
Step 5: Calculate the final position of the ship from the port.
- Add the changes in the y and x coordinates from both steps together.
- The final y-coordinate is the sum of the changes in y from the first and second legs.
- The final x-coordinate is the sum of the changes in x from the first and second legs.
Step 6: Find the bearing of the final position from the port.
- Calculate the angle between the final position and the original position of the port from a right triangle formed by the change in coordinates.
- Use the atan2 function or reference a trigonometric table to find the angle.
- Convert the angle to a compass bearing by subtracting it from 360 if necessary.
- This bearing will be the answer to your question.
By following these steps, you should be able to obtain the final bearing of the ship from the port after traveling 7 miles with a bearing of S 40 W and then 11 miles with a bearing of N 50 W.
To find the bearing of the ship from the port at the given time, we can use the concept of vector addition.
1. Start by drawing a diagram to visualize the ship's movements. Place the starting point of the ship (port) as the origin of the diagram.
2. From the starting point, draw a line segment with a bearing of S 40 W for 7 miles. Denote the endpoint of this segment as Point A.
3. From Point A, draw a second line segment with a bearing of N 50 W for 11 miles. Denote the endpoint of this segment as Point B.
4. To find the bearing of the ship from the port at this time, we need to determine the angle between the line segment AB and the north-south reference line (vertical line).
5. Draw a line down from Point B to the north-south reference line and denote the intersection point as Point C.
6. Now, we have a triangle formed by the line segments AB, BC, and AC.
7. We can calculate the angle CAB using trigonometry. Since we know the lengths of AB and BC, we can use the tangent function.
tan(CAB) = BC / AB
tan(CAB) = 11 / 7
8. Use the inverse tangent function (arctan) to find the angle CAB:
CAB = arctan(11 / 7)
9. Finally, to find the bearing of the ship from the port, subtract the angle CAB from 180 degrees (since it is measured clockwise from the north), and round the result to the nearest degree.
10. The final bearing of the ship from the port is: S 23 E.