A hiker begins a trip by first walking 25.0 km 45.0° south of east from her base camp.On the second day she walks 40.0 km in a direction 60.0° north of east, at which point she discovers a forest ranger’s tower.Find the magnitude of the displacement from base camp.

X=(25 COS 45) + (40 SIN 60)
Y= (-25 SIN 45) + (40 COS 60)

To find the magnitude of the displacement from the base camp, we need to calculate the components in the x and y directions and then find the magnitude of the resulting vector.

The x-component of the displacement can be calculated as follows:
X = (25 km * cos(45°)) + (40 km * sin(60°))
X = (25 km * 0.7071) + (40 km * 0.866)
X = 17.678 km + 34.64 km
X = 52.318 km

The y-component of the displacement can be calculated as follows:
Y = (-25 km * sin(45°)) + (40 km * cos(60°))
Y = (-25 km * 0.7071) + (40 km * 0.5)
Y = - 17.678 km + 20 km
Y = 2.322 km

Now, we can find the magnitude of the displacement using the Pythagorean theorem:
Magnitude = sqrt(X^2 + Y^2)
Magnitude = sqrt((52.318 km)^2 + (2.322 km)^2)
Magnitude = sqrt(2741.252 km^2 + 5.388 km^2)
Magnitude = sqrt(2746.64 km^2)
Magnitude ≈ 52.43 km

Therefore, the magnitude of the displacement from the base camp is approximately 52.43 km.

To find the magnitude of the displacement from the base camp, we need to find the x and y components of the displacement.

We can break down the first day's walk into horizontal (x) and vertical (y) components. The hiker walks 25.0 km in a direction 45.0° south of east.

The x-component is given by:
X = 25 * cos(45°)

The y-component is given by:
Y = -25 * sin(45°)

Next, we need to calculate the second day's walk. The hiker walks 40.0 km in a direction 60.0° north of east.

The x-component is given by:
X = X + 40 * sin(60°)

The y-component is given by:
Y = Y + 40 * cos(60°)

To find the magnitude of the displacement, we can use the Pythagorean theorem:
Magnitude of displacement = √(X² + Y²)

Substituting the calculated values, we have:
Magnitude of displacement = √((25 * cos(45°) + 40 * sin(60°))² + (-25 * sin(45°) + 40 * cos(60°))²)

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

so

the magnitude is sqrt(X^2+Y^2)

38 km