The captain of a ship sailing along a bearing of 070oT observes two lighthouses in a line due North of himself. The ship continues on the same course for a further 5 km. Now, one of the lighthouses has a bearing of 280oT from the ship and the other has a bearing of 320oT from the ship. Calculate the distance between the lighthouses.
draw a diagram. If the two lighthouses are at S and N, and the ship moved from A to B, then let the east-west distance moved by the ship be x.
x = 5cos20°
SN = x tan50° - x tan10°
To calculate the distance between the lighthouses, we can use trigonometry. Here are the steps to solve the problem:
Step 1: Understand the problem
The captain observes two lighthouses in a line due North of himself. After sailing for 5 km, one lighthouse is at a bearing of 280oT from the ship, and the other lighthouse is at a bearing of 320oT from the ship. We need to calculate the distance between the lighthouses.
Step 2: Draw a Diagram
To visualize the problem, let's draw a diagram. Place the captain's ship at the starting point, and draw a line due North to represent the position of the lighthouses. Label the first lighthouse as A and the second one as B. Mark the position of the ship after sailing 5 km.
N
|
| A --------> lighthouse 1 (280oT)
| /
| /
| /
Ship --------- B --------> lighthouse 2 (320oT)
Step 3: Determine Angles
Given the bearings of the lighthouses from the ship, we can find the angles of triangles by subtracting the bearing from 360oT (to convert from T - True bearing to N - North bearing). In this case, the angle at the first lighthouse A is 360oT - 280oT = 80o, and the angle at the second lighthouse B is 360oT - 320oT = 40o.
Step 4: Apply Trigonometry
Since we have a right-angled triangle formed between the ship and each lighthouse, we can use basic trigonometry to find the distances. In this case, we know the ship has traveled 5 km. Let's label the distances from the ship to the lighthouses as a and b.
For the triangle with lighthouse A:
tan(80o) = a / 5
For the triangle with lighthouse B:
tan(40o) = b / 5
Solve these two equations to find the values of a and b.
Step 5: Calculate the Distance
Now that we have the values of a and b, we can use the Pythagorean theorem to find the distance between the lighthouses.
Distance = √(a^2 + b^2)
Plug in the values of a and b and calculate the distance between the lighthouses.
By following these steps, you should be able to calculate the distance between the lighthouses based on the given information.