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.

41.32

To find the magnitude of the displacement from the base camp, we can break down the hiker's movements into eastward and northward components.

On the first day, the hiker walks 25.0 km at an angle of 45.0° south of east.
The eastward displacement is given by:
east_displacement_day1 = distance * cos(angle)
east_displacement_day1 = 25.0 km * cos(45.0°) ≈ 17.68 km (rounded to two decimal places)

The northward displacement is given by:
north_displacement_day1 = distance * sin(angle)
north_displacement_day1 = 25.0 km * sin(45.0°) ≈ 17.68 km (rounded to two decimal places)

On the second day, the hiker walks 40.0 km at an angle of 60.0° north of east.
The eastward displacement is given by:
east_displacement_day2 = distance * cos(angle)
east_displacement_day2 = 40.0 km * cos(60.0°) ≈ 20.00 km (rounded to two decimal places)

The northward displacement is given by:
north_displacement_day2 = distance * sin(angle)
north_displacement_day2 = 40.0 km * sin(60.0°) ≈ 34.64 km (rounded to two decimal places)

To find the total displacement, we need to sum the eastward and northward displacements:
total_east_displacement = east_displacement_day1 + east_displacement_day2 ≈ 17.68 km + 20.00 km ≈ 37.68 km (rounded to two decimal places)
total_north_displacement = north_displacement_day1 + north_displacement_day2 ≈ 17.68 km + 34.64 km ≈ 52.32 km (rounded to two decimal places)

Now, we can find the magnitude of the displacement using the Pythagorean theorem:
magnitude_of_displacement = sqrt((total_east_displacement)^2 + (total_north_displacement)^2)
magnitude_of_displacement = sqrt((37.68 km)^2 + (52.32 km)^2) ≈ 63.85 km (rounded to two decimal places)

Therefore, the magnitude of the displacement from the base camp is approximately 63.85 km.

To find the magnitude of the displacement from the base camp, we need to calculate the total vector sum of the two displacements.

The first displacement is given as 25.0 km at an angle of 45.0° south of east. We can represent this as a vector with its components in the east and south directions. Using trigonometry, we can calculate the east and south components of this displacement as follows:

East component = 25.0 km * cos(45.0°) = 17.68 km
South component = 25.0 km * sin(45.0°) = 17.68 km (since the south direction is opposite to the north direction)

The second displacement is given as 40.0 km at an angle of 60.0° north of east. Similarly, we can calculate its components:

East component = 40.0 km * cos(60.0°) = 20.0 km
North component = 40.0 km * sin(60.0°) = 34.64 km

Now, we can add the east, south, and north components to find the overall displacement:

East displacement = 17.68 km + 20.0 km = 37.68 km (east)
North displacement = 34.64 km (north)

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

Magnitude of displacement = √(East displacement^2 + North displacement^2)
Magnitude of displacement = √(37.68 km^2 + 34.64 km^2)
Magnitude of displacement = √(1418.9824 km^2 + 1199.7696 km^2)
Magnitude of displacement ≈ √(2618.752 km^2)
Magnitude of displacement ≈ 51.17 km

Therefore, the magnitude of the displacement from the base camp is approximately 51.17 km.

You can draw a figure, and use the law of cosine/sines or..

figure components then add.
first day:
N: 25cos135 N (angles from 000 clockwise)
E: 2sin135 E

second day
N: 40cos30 N
E: 40sin30 E

Now add components:
N: 40cos30 + 25cos135
E: 40sin30 + 25sin135

Now, displacement..
d=sqrt(N^2 + E^2)