In wandering, a grizzly bear makes a displacement of 1588 m due west, followed by a displacement of 3390 m in a direction 38.0° north of west. What are the magnitude and the direction of the displacement needed for the bear to return to its starting point? Specify the direction relative to due east.

 

To find the magnitude and direction of the displacement needed for the bear to return to its starting point, we need to find the resultant displacement by adding the two given displacements together.

First, let's break down the given displacements into their x (east-west) and y (north-south) components.

For the displacement of 1588 m due west:
- x-component: -1588 m (negative because it is westward)
- y-component: 0 m (no north-south movement)

For the displacement of 3390 m at a direction 38.0° north of west:
- x-component: -3390 m * cos(38.0°) (negative because it is westward)
- y-component: 3390 m * sin(38.0°) (positive because it is northward)

Now, let's calculate the components:

x-component = -3390 m * cos(38.0°)
= -3390 m * 0.7880108
= -2672.17 m

y-component = 3390 m * sin(38.0°)
= 3390 m * 0.6156615
= 2085.75 m

To find the resultant displacement, we add the x-components and the y-components:

x-component of resultant = -1588 m + (-2672.17 m)
= -4260.17 m

y-component of resultant = 0 m + 2085.75 m
= 2085.75 m

Now, we can find the magnitude and direction of the resultant displacement.

Magnitude:
magnitude = sqrt((x-component of resultant)^2 + (y-component of resultant)^2)
= sqrt((-4260.17 m)^2 + (2085.75 m)^2)
= sqrt(18140760.92 + 4350166.61)
= sqrt(22490927.53)
= 4743.65 m

Direction:
direction = arctan((y-component of resultant) / (x-component of resultant))
= arctan(2085.75 m / (-4260.17 m))
= arctan(-0.4895927)
= -25.0°

Since the question asks for the direction relative to due east, we can convert the direction angle to an eastward angle by subtracting it from 180°:

direction relative to due east = 180° - 25.0°
= 155.0°

Therefore, the magnitude of the displacement needed for the bear to return to its starting point is 4743.65 m, and the direction relative to due east is 155.0°.