From a ship, two lighthouses can be seen bearing N 40 degrees E. After the ship sails at 15 knots on a course of 135 degrees for an hour and 20 mins, the two lighthouses now has a bearing of 10 degrees and 345 degrees.
a) Find the distance of the ship from the latter position to the farther lighthouse?
b) Find the distance between the two lighthouses? .... thank you in advance.
The ship travels 20 miles in the time given.
If we label the points S for ship (current position), N and F for Near and Far lighthouses, then if we call the original position O,
In triangle SNF,
S=25°
N=125°
F=30°
In triangle ONS,
n = 20
N=55°
O=95°
so,
NS/sin95° = 20/sin55°
We want SF and NF
Now you can use the law of sines in NSF to find the other two sides.
To solve this problem, we can use the concept of relative motion and trigonometry. Let's break down the steps to find the answers to the given questions:
a) Find the distance of the ship from the latter position to the farther lighthouse:
Step 1: Convert the time of travel from hours and minutes to decimal form.
1 hour + 20 minutes = 1.33 hours (since 20 minutes is 1/3 of 1 hour).
Step 2: Calculate the distance covered by the ship using the formula:
Distance = Speed x Time
Since the ship is sailing at a speed of 15 knots and traveled for 1.33 hours:
Distance = 15 knots x 1.33 hours = 19.95 nautical miles (approximately 20 nautical miles).
Step 3: Determine the change in bearing of the lighthouses after the ship sailed.
Initially, the bearing of the two lighthouses was N 40 degrees E.
After sailing for 1 hour and 20 minutes on a course of 135 degrees, the new bearing of the lighthouses is 10 degrees and 345 degrees.
Step 4: Calculate the bearing change by subtracting the initial bearing from the new bearing.
Bearing change for the first lighthouse = 10 degrees - 40 degrees = -30 degrees
Bearing change for the second lighthouse = 345 degrees - 40 degrees = 305 degrees
Step 5: Convert the bearing change to the number of degrees covered by the ship.
Since the ship sailed for 1.33 hours (80 minutes):
Angle covered = Bearing change x Time
For the first lighthouse: -30 degrees x 1.33 hours = -39.9 degrees (approximately -40 degrees)
For the second lighthouse: 305 degrees x 1.33 hours = 406.65 degrees (approximately 407 degrees)
Step 6: Convert the distance and angle covered by the ship to rectangular coordinates.
Distance covered (d) = 20 nautical miles (from step 2)
Angle covered in rectangular coordinates (θ) = 407 degrees (from step 5)
Convert θ to radians:
θ (in radians) = θ (in degrees) × π / 180
θ (in radians) = 407 degrees × π / 180 = 7.112 radians (approximately)
Convert d to rectangular coordinates:
x = d × cos(θ)
y = d × sin(θ)
x = 20 × cos(7.112) = 19.958 nautical miles (approximately 20 nautical miles)
y = 20 × sin(7.112) = 2.466 nautical miles (approximately 2.5 nautical miles)
Step 7: Calculate the distance between the latter position of the ship and the farther lighthouse using the Pythagorean theorem.
Distance = √(x^2 + y^2)
Distance = √((19.958)^2 + (2.466)^2) = √(398.321 + 6.078) = √404.399 = 20.109 nautical miles (approximately 20 nautical miles).
Therefore, the distance of the ship from the latter position to the farther lighthouse is approximately 20 nautical miles.
b) Find the distance between the two lighthouses:
Since we have the bearings of the two lighthouses, we can use trigonometry to find the distance between them.
Step 1: Calculate the angle between the bearing lines of the two lighthouses.
Angle between bearings = 345 degrees - 10 degrees = 335 degrees.
Step 2: Convert the angle between bearings to radians.
Angle between bearings (in radians) = 335 degrees × π / 180 = 5.847 radians (approximately).
Step 3: Use the Law of Cosines to find the distance between the two lighthouses.
Distance = √(d1^2 + d2^2 - 2 * d1 * d2 * cos(angle))
Distance = √(20^2 + 20^2 - 2 * 20 * 20 * cos(5.847))
Distance = √(400 + 400 - 800 * cos(5.847))
Distance = √(800 - 800 * cos(5.847))
Distance = √(800 - 800 * 0.996) = √(800 - 796.8) = √3.2 = 1.788 nautical miles (approximately 1.8 nautical miles).
Therefore, the distance between the two lighthouses is approximately 1.8 nautical miles.