A ship sails for 5 hours from position A: (27° 38' N, 112° 45' W) at a speed of 16 knots on a course of 295°T.
Determine the course angle.
To determine the course angle, we need to find the initial and final positions of the ship.
The initial position is given as coordinates (27° 38' N, 112° 45' W).
We convert these coordinates to decimal degrees.
For latitude, 1 degree is equal to 60 minutes, so 27° 38' N is equal to 27 + (38/60) = 27.6333 degrees N.
For longitude, 1 degree is equal to 60 minutes, so 112° 45' W is equal to -112 - (45/60) = -112.75 degrees W.
The final position can be calculated by multiplying the speed of the ship (16 knots) by the time sailed (5 hours). This gives us a distance of 16 knots/hour * 5 hours = 80 nautical miles.
Now, we need to consider the course angle of 295°T. T stands for true heading, which means the angle is measured clockwise from true north. However, we need to convert this to a compass bearing where the angle is measured clockwise from magnetic north.
The difference between true north and magnetic north, known as magnetic variation, is different for different locations. For the given location, let's assume the magnetic variation is -5 degrees (meaning magnetic north is 5 degrees west of true north).
To convert the course angle from true heading to magnetic bearing, we add the magnetic variation (-5 degrees) to the course angle (295 degrees). This gives us a magnetic bearing of 295 + (-5) = 290 degrees.
Therefore, the course angle is 290 degrees.
To determine the course angle, we need to use the information given. The course angle is the angle between the direction the ship is sailing and the reference north.
First, we need to convert the coordinates from degrees, minutes, and seconds to decimal degrees.
The coordinates of position A in decimal degrees are:
Latitude: 27 + 38/60 = 27.6333° N
Longitude: -112 - 45/60 = -112.7500° W
Next, we need to find the change in latitude and longitude after sailing for 5 hours at a speed of 16 knots on a course of 295°T.
Change in Latitude:
The distance traveled in 5 hours at a speed of 16 knots can be calculated by multiplying the speed by the time: 16 knots * 5 hours = 80 nautical miles.
1 nautical mile is equal to 1 minute of latitude, so the change in latitude is 80 minutes.
Change in Longitude:
To find the change in longitude, we need to calculate the distance traveled in terms of latitude degrees. Since the earth is not a flat surface, the distance traveled in terms of longitude degrees will vary based on the latitude. We can use the formula:
Change in Longitude = Distance Traveled / (Cosine(Latitude) * Conversion Factor)
The conversion factor is obtained by multiplying the equatorial circumference of the earth (21,639.7 nm) by the cosine of the latitude.
Using the given latitude of 27.6333° N, we can calculate the change in longitude:
Change in Longitude = 80 / (Cosine(27.6333) * (21639.7 * Cosine(27.6333))) = 0.9057°
Now, we can calculate the final position by adding the change in latitude and longitude to the initial position:
Latitude: 27.6333 + 80/60 = 28.3333° N
Longitude: -112.7500 + 0.9057 = -111.8443° W
To determine the course angle, we need to find the bearing between the initial and final positions.
Using the formula: Bearing = arctan((sin(change in longitude) * cos(final latitude)) / (cos(initial latitude) * sin(final latitude) - sin(initial latitude) * cos(final latitude)))
Bearing = arctan((sin(0.9057) * cos(28.3333)) / (cos(27.6333) * sin(28.3333) - sin(27.6333) * cos(28.3333)))
Bearing ≈ 294.473°
Therefore, the course angle is approximately 294.473°.