a ship leaves port at 12:00 noon sailing at a bearing 193° measured clockwise from north. if the ship sails 20 knots, how many nautical miles south and how many nautical miles west will the ship have traveled by 6:00pm?
the bearing is 13º west of south
20 knots for 6 hr is 120 nautical miles
south = 120 cos(13º)
west = 120 sin(13º)
To find out how many nautical miles the ship will have traveled by 6:00 pm, we need to determine the distance traveled in the southward and westward directions.
Since the ship is sailing at a bearing of 193° measured clockwise from north, we can break down the bearing as follows:
- 193° clockwise from north is equivalent to 360° - 193° counterclockwise from north.
- 360° - 193° = 167° counterclockwise from north.
Now, we can calculate the distances traveled in the southward and westward directions separately.
1. Distance traveled southward:
- To find the distance traveled southward, we need to determine the component of the ship's speed in the southward direction.
- The component of the ship's speed in the southward direction is given by the formula: Speed * cos(angle).
- In this case, the speed is 20 knots, and the angle is 167° counterclockwise from north.
- So, the distance traveled southward can be calculated as: Distance southward = 20 knots * cos(167°).
2. Distance traveled westward:
- To find the distance traveled westward, we need to determine the component of the ship's speed in the westward direction.
- The component of the ship's speed in the westward direction is given by the formula: Speed * sin(angle).
- In this case, the speed is 20 knots, and the angle is 167° counterclockwise from north.
- So, the distance traveled westward can be calculated as: Distance westward = 20 knots * sin(167°).
By substituting the values into the above formulas, we can calculate the distances traveled in the southward and westward directions.
To determine how many nautical miles south and west the ship would have traveled by 6:00 PM, we need to calculate the distance traveled in each direction separately.
First, let's calculate the distance traveled south.
Speed is measured in knots, which is equivalent to nautical miles per hour (nm/h). Since the ship sails at a speed of 20 knots, it means that the ship will travel 20 nautical miles in one hour. Therefore, between 12:00 PM and 6:00 PM, there are 6 hours.
Distance traveled south = Speed x Time
Distance traveled south = 20 knots/hour x 6 hours
Distance traveled south = 120 nautical miles south
Now, let's calculate the distance traveled west.
The bearing of 193° measured clockwise from north means that the ship is sailing in the southwest direction. To calculate the distance traveled west, we need to use trigonometry.
Given:
Bearing angle = 193°
Speed = 20 knots
Time = 6 hours
To calculate the distance traveled west, we can use the cosine of the bearing angle and multiply it by the speed and time:
Distance traveled west = Speed x Time x cos(Bearing)
Distance traveled west = 20 knots/hour x 6 hours x cos(193°)
Using a calculator, cos(193°) ≈ -0.978
Distance traveled west ≈ 20 knots/hour x 6 hours x -0.978
Distance traveled west ≈ -117.36 nautical miles west
Note: The negative sign indicates that the ship traveled westward.
So, by 6:00 PM, the ship will have traveled approximately 120 nautical miles south and 117.36 nautical miles west.