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?

To solve this problem, we can use trigonometry and basic geometry principles. Let's analyze the given information step by step:

a) Find the distance of the ship from the latter position to the farther lighthouse:

1. Start by drawing a diagram to visualize the situation. Draw a triangle to represent the ship's initial and final positions, with the lighthouses at the vertices.

- Let's label the initial position of the ship as A, the final position as B, and the two lighthouses as L1 and L2.

- We know that the bearing N 40 degrees E from the initial position of the ship intersects with the line joining L1 and L2. So, we can draw a line from A towards N 40 degrees E that intersects with the line between L1 and L2.

- Similarly, the bearing of 10 degrees from the final position of the ship intersects with the line joining L1 and L2. So, we can draw a line from B towards 10 degrees that intersects with the line between L1 and L2.

2. Now, let's find the bearing angles at the initial and final positions of the ship.

- The initial bearing N 40 degrees E represents the angle between the direction North and the line joining the initial position of the ship to the intersection point on the line between L1 and L2.

- Similarly, the final bearing of 10 degrees represents the angle between the direction North and the line joining the final position of the ship to the intersection point on the line between L1 and L2.

3. Calculate the change in bearing angle between the initial and final positions of the ship.

- Subtract the initial bearing (40 degrees) from the final bearing (10 degrees) to find the change in angle. The result will be the angle made by the line between the initial and final positions of the ship with the line between L1 and L2.

4. Use the change in bearing angle and the sailing time of 1 hour and 20 mins (or 1.33 hours) to calculate the distance traveled by the ship.

- The ship sails at a constant speed of 15 knots (which is equivalent to 15 nautical miles per hour).

- Multiply the sailing time (1.33 hours) by the ship's speed (15 knots) to get the distance traveled by the ship.

5. With the distance traveled by the ship and the change in bearing angle, we can now find the distance from the latter position of the ship to the farther lighthouse.

- Use the law of sines to calculate the distance from the intersection point on the line between L1 and L2 to the farther lighthouse.

- The law of sines states that the ratio of the length of a side of a triangle to the sine of the angle opposite that side is constant.

- Apply the law of sines to the triangle formed with the intersection point on the line between L1 and L2, the farther lighthouse, and the position of the ship at B.

b) Find the distance between the two lighthouses:

- By using the distances from both lighthouses to the ship's final position (found in part a), you can determine the distance between the two lighthouses by subtracting the distance from the nearer lighthouse from the distance from the farther lighthouse.

Performing these calculations and steps should give you the answers to both questions (a) and (b). Alternatively, you can use online tools or software that can perform these calculations, taking into account the necessary formulas and trigonometric principles.