A hiker begins a trip by first walking 25.0 km southeast from her base camp. On the 2nd day, she walks 40.0 km in a direction 600 north of east, at which point she discovers a forest rangers tower. Find the magnitude and direction of displacement from base camp.

To find the magnitude and direction of displacement from the base camp, we need to use vector addition.

First, let's break down the given information into vectors:

1. The hiker walks 25.0 km southeast from the base camp. This can be represented as a vector with a magnitude of 25.0 km and a direction of 135° (southeast is halfway between south and east).

2. On the 2nd day, the hiker walks 40.0 km in a direction 600 north of east. To represent this vector, we need to convert the angle 60° north of east into a standard mathematical angle. Since the north direction is clockwise from the standard positive y-axis, we subtract the angle from 90°. Therefore, the angle is 30° in the positive x-axis direction. The magnitude of this displacement is 40.0 km.

To find the total displacement vector, we add these two vectors together using vector addition. However, we need to convert the angles into Cartesian coordinates to make the addition easier.

The southeast vector can be represented in Cartesian coordinates as (25.0 km * cos(135°), 25.0 km * sin(135°)).

The north-east vector can be represented in Cartesian coordinates as (40.0 km * cos(30°), 40.0 km * sin(30°)).

Now we can perform vector addition:

X component = (25.0 km * cos(135°)) + (40.0 km * cos(30°))
Y component = (25.0 km * sin(135°)) + (40.0 km * sin(30°))

To find the magnitude of the displacement, we use the Pythagorean theorem:

Magnitude = sqrt((X component)^2 + (Y component)^2)

Finally, we can find the direction of the displacement using the inverse tangent function:

Direction = atan(Y component / X component)

Calculating these values will give us the magnitude and direction of the displacement from the base camp.