A hiker leaves her camp and walks 3.5 km in a direction of 55° south of west to the lake. After a short rest at the lake, she hikes 2.7 km in a direction of 16° east of south to the scenic overlook.
To find the displacement of the hiker, we can break down the given information into vectors and then combine them.
First, let's analyze the first part of the hike:
Distance: 3.5 km
Direction: 55° south of west
To convert this direction to a vector, we need to split it into its components.
Component west:
3.5 km * cos(55°)
Component south:
3.5 km * sin(55°) * (-1) [since it is south]
Thus, the vector for the first part of the hike is:
Vector_1 = (3.5 km * cos(55°), -3.5 km * sin(55°))
Now let's analyze the second part of the hike:
Distance: 2.7 km
Direction: 16° east of south
Again, we need to convert this direction to a vector.
Component south:
2.7 km * cos(16°) * (-1) [since it is south]
Component east:
2.7 km * sin(16°)
Thus, the vector for the second part of the hike is:
Vector_2 = (-2.7 km * cos(16°), 2.7 km * sin(16°))
To find the total displacement, we simply add the two vectors together:
Total displacement = Vector_1 + Vector_2
Plugging in the values, we get:
Total displacement = (3.5 km * cos(55°) - 2.7 km * cos(16°), -3.5 km * sin(55°) - 2.7 km * sin(16°))
To find the magnitude of the displacement, we can use the Pythagorean theorem:
Magnitude = sqrt[(3.5 km * cos(55°) - 2.7 km * cos(16°))^2 + (-3.5 km * sin(55°) - 2.7 km * sin(16°))^2]