An aeroplane flies due north for 200 km and then turns due east and flies for a further distance of 325km. Find the bearing,to the nearest degree,of the aeroplane from its starting point.
335km
To find the bearing of the aeroplane from its starting point, we need to use trigonometry and the concept of vectors.
Step 1: Draw a diagram representing the situation. Place the starting point (which can be considered as the origin of a coordinate system) at the center of the diagram. Label the distance flown north as 200 km and the distance flown east as 325 km.
Step 2: Find the resultant vector by adding the north and east vectors. In this case, the resultant vector represents the displacement of the aeroplane from its starting point.
To add the north and east vectors, we need to convert them from polar coordinates (length and bearing) to rectangular coordinates (x and y components).
- The north vector:
The length of the north vector is 200 km, and it is in the direction of the positive y-axis. Therefore, its rectangular coordinates are (0, 200).
- The east vector:
The length of the east vector is 325 km, and it is in the direction of the positive x-axis. Therefore, its rectangular coordinates are (325, 0).
Step 3: Add the x and y components of the north and east vectors to obtain the components of the resultant vector.
Adding the x components: 0 + 325 = 325
Adding the y components: 200 + 0 = 200
The components of the resultant vector are (325, 200) in the rectangular coordinate system.
Step 4: Calculate the magnitude and bearing of the resultant vector.
The magnitude (r) of the resultant vector is found using the Pythagorean theorem: r = √(x^2 + y^2)
r = √(325^2 + 200^2) ≈ 382.61 km
The angle (θ) of the resultant vector with the positive x-axis is found using the arctan function: θ = arctan(y/x)
θ = arctan(200/325) ≈ 32.92°
Step 5: Determine the bearing of the aeroplane from its starting point.
Since the aeroplane started flying due north, we need to add 90° (turn clockwise) to the angle θ to get the bearing with respect to the north.
Bearing = 32.92° + 90° ≈ 122.92°
Therefore, the bearing of the aeroplane from its starting point is approximately 122.92°, rounded to the nearest degree.