a ship sails from a point at a distance of 4 km and a bearing of 40 degree noth of east relative to a lighthouse to a point 6 km from the lighthouse at 60 degree north to west.
calculate
a) the displacement
b)the least distance between them
To calculate the displacement and the least distance between the ship and the lighthouse, we can first draw a diagram to represent the situation.
Let's assume the position of the ship as A, the lighthouse as L, and the final destination of the ship as B.
a) Calculating the displacement:
1. Draw a straight line to represent the initial bearing of the ship relative to the lighthouse. This line will have a length of 4 km and a direction of 40 degrees north of east. Label this line as AL.
2. From point L, draw another line in the direction of 60 degrees north to west. This line will have a length of 6 km. Label this line as LB.
3. Complete the triangle by drawing a line connecting points A and B.
4. The line segment AB represents the displacement of the ship.
5. Distance and direction can be calculated using trigonometry: We can use the Law of Cosines to find the length of line segment AB and the Law of Sines to find the angle of line segment AB.
b) Calculating the least distance between the ship and the lighthouse:
1. Draw a line from point A perpendicular to line LB. This line segment represents the shortest distance between the ship and the lighthouse.
2. To find this shortest distance, we can use trigonometry again. The length of line segment AC can be calculated using the trigonometric functions (sine and cosine).
With the given information, we can calculate the displacement and the least distance between the ship and the lighthouse using trigonometric formulas and basic geometry principles.