A ship leaves port at noon and has a bearing of S 25° W. The ship sails at 15 knots. How many nautical miles south and how many nautical miles west does the ship travel by 6:00 P.M.? (Round your answers to two decimal places.)
Miles South?
Miles West?
distance in the direction of the bearing = 6(15) or 90 n miles.
make your sketch to see:
sin 25° = x/90
x = 90sin25 = appr 38.04 n miles
cos25= y/90
y = 90cos25= 81.57 n miles
To find the distance traveled south and west by the ship, we can use the formula:
Distance = Speed * Time
Given that the ship sails at 15 knots and the time is from noon to 6:00 P.M., which is a total of 6 hours, we can calculate the distance traveled as follows:
Distance South = Speed * Time
Distance South = 15 knots * 6 hours
Distance South = 90 nautical miles
Therefore, the ship travels 90 nautical miles south.
To calculate the distance traveled west, we need to use trigonometry. Since the ship has a bearing of S 25° W, it is moving south and veering to the west by 25° from the south direction. We can use the sine and cosine functions to determine the distances:
Distance West = Speed * Time * cosine(angle)
Angle = 25°
Speed = 15 knots
Time = 6 hours
Distance West = 15 knots * 6 hours * cosine(25°)
Distance West ≈ 15 knots * 6 hours * 0.90631
Distance West ≈ 81.956 nautical miles
Therefore, the ship travels approximately 81.96 nautical miles west.
To find the miles south and miles west traveled by the ship, we can use the basic trigonometric relationships of sine and cosine.
Given:
Bearing = S 25° W
Speed = 15 knots
Duration = 6 hours
Step 1: Convert the bearing to angles that work with trigonometric functions.
The bearing S 25° W can be converted to S (180° - 25°) W = S 155° W.
Step 2: Break down the bearing into north-south and east-west components.
The angle of S 155° W can be broken down into an angle of 25° south of due west and 90° - 25° = 65° west of due north.
Step 3: Use the trigonometric relationships to find the distances traveled.
The distance south can be found using sine:
Miles south = Speed × time × sin(angle south)
Miles south = 15 knots × 6 hours × sin(25°)
Miles south ≈ 30.6 nautical miles (rounded to two decimal places)
The distance west can be found using cosine:
Miles west = Speed × time × cos(angle west)
Miles west = 15 knots × 6 hours × cos(65°)
Miles west ≈ 42.89 nautical miles (rounded to two decimal places)
Therefore, the ship traveled approximately 30.6 nautical miles south and 42.89 nautical miles west by 6:00 P.M.