Main Answer
Part A: The geese are 4.0 km - x west of their initial position.
Part B: The magnitude of their displacement is approximately 6.5 km.
Explanation
Part A: To determine how far west the geese are of their initial position, we need to find the horizontal component of their displacement during the first leg of their journey. Since they fly due west for a distance of 4.0 km, this component is simply 4.0 km.
However, during the second leg of their journey, they fly at an angle of 30 degrees north of west. To find the horizontal component of this displacement, we can use trigonometry to calculate the x-coordinate of their final position relative to their initial position:x = 4.5 km cos(30 degrees) = 3.9 km.
Now, we can subtract this value from the horizontal component of their first leg to get the total horizontal displacement:Total Horizontal Displacement = 4.0 km - 3.9 km = 0.1 km (rounded to two decimal places).
Since this value is positive, it means that they are still west of their initial position by approximately 100 m (assuming that their initial position was at the origin).
If we wanted to be more precise, we could calculate the exact coordinates of their initial and final positions using a map or GPS device, and then find the distance between them using a formula for great-circle distances on a sphere (which takes into account the curvature of the Earth).
Part B: To find the magnitude of their displacement, we can use the distance formula to calculate the straight-line distance between their initial and final positions:
Displacement Magnitude= approximately 4.5 km (rounded to two decimal places).This value represents the total distance traveled by the geese from their initial position to their final position, regardless of whether they traveled in a straight line or not.