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 the concept of vectors.
a) To find the distance of the ship from the latter position to the farther lighthouse, we need to break down the given information. Initially, the two lighthouses were bearing N 40 degrees E when observed from the ship. After the ship sailed for 1 hour and 20 minutes (or 1.33 hours) at a speed of 15 knots on a course of 135 degrees, the new bearing of the lighthouses is given as 10 degrees and 345 degrees.
First, let's calculate the distance covered by the ship using the formula:
Distance = Speed * Time
Distance = 15 knots * 1.33 hours
Next, we need to convert the distance covered by the ship into a vector form. To do this, we need to consider the direction of the ship's course. The course is 135 degrees, which means it is directed towards the southeast.
Now, let's calculate the vector components of the ship's course:
x-component = Distance * cos(angle)
y-component = Distance * sin(angle)
Using the above formulas, we can find the x and y components of the distance covered by the ship.
Once we have the vector components, we need to add them to the original position of the ship to find its new position. This can be done by adding the x-components and y-components of both positions.
After finding the new position of the ship, we need to calculate the bearing of the latter lighthouse from the ship's new position. This can be done by finding the angle between the ship's new position and the latter lighthouse using trigonometry.
Finally, we can use the bearing and the ship's new position to calculate the distance between the ship and the latter lighthouse.
b) To find the distance between the two lighthouses, we can use the distance formula in coordinate geometry. We already know the coordinates of both lighthouses, so plugging in those values into the formula will give us the distance between them.
Distance = sqrt((x2 - x1)^2 + (y2 - y1)^2)
By substituting the values of the lighthouse coordinates into the formula, we can find the distance between the two lighthouses.