If a ship leaves port at 9:00 a.m. and sails due north for 3 hours at 12 knots, then turns N 30° E for another hour, how far from port is the ship?
If you make your sketch you will see a cosine law problem.
let the distance from port be k nautical miles
d^2 = 36^2 + 12^2 - 2(12)(36)cos 150°
= ...
d = √...
= ....
To find the distance from the port, we need to break down the ship's journey into two components: the distance traveled during the first 3 hours and the distance traveled during the next hour. Let's calculate it step by step:
Step 1: Calculate the distance traveled in the first 3 hours.
Since the ship is sailing at a constant speed of 12 knots for 3 hours, we can use the formula: distance = speed × time.
The distance traveled in the first 3 hours is: distance_1 = speed × time = 12 knots × 3 hours.
Thus, the distance traveled in the first 3 hours is 36 nautical miles (since 1 knot is equal to 1 nautical mile per hour).
Step 2: Calculate the distance traveled during the next hour.
The ship turns N 30° E for another hour after the first 3 hours of sailing due north.
To calculate this distance, we need to determine how far the ship travels east (N 30° E) and north during this hour.
Using trigonometry, we can find the east and north components of the ship's movement. Let's call the distance traveled east as "distance_east" and the distance traveled north as "distance_north."
From the given information, we can see that we have a right-angled triangle formed by the east, north, and hypotenuse sides. The angle between the east side and the hypotenuse is 30° (N 30° E).
Using trigonometry, we can determine the distance_east and distance_north using the formulae:
distance_east = hypotenuse × cos(angle)
distance_north = hypotenuse × sin(angle)
Since the ship sailed north for 3 hours at 12 knots, the hypotenuse is the distance traveled in the first 3 hours (36 nautical miles).
Plugging in the values, we have:
distance_east = 36 nautical miles × cos(30°)
distance_north = 36 nautical miles × sin(30°)
Calculating these values, we find that distance_east is approximately 31.18 nautical miles and distance_north is approximately 18 nautical miles.
Step 3: Calculate the total distance from the port.
To find the distance from the port, we need to calculate the resultant distance traveled. This can be done using the Pythagorean theorem:
resultant_distance = √(distance_traveled_1^2 + distance_east^2 + distance_north^2)
Plugging in the values:
resultant_distance = √(36^2 + 31.18^2 + 18^2)
resultant_distance = √(1296 + 970.3924 + 324)
resultant_distance = √(2590.3924 + 324)
resultant_distance = √(2914.3924)
resultant_distance is approximately equal to 53.96 nautical miles.
Therefore, the ship is approximately 53.96 nautical miles from the port.