Final answer:
The person ends up 5.7 miles east and 12.78 miles south of the starting point after adding up the vectors representing each segment of their walk. We translate the southeast and southwest directions into their respective components along the x and y axes before summing them up.
Step-by-step explanation:
To solve the student's problem, we should break down the various movements into their constituent vectors and then add these vectors together to find the resulting displacement. We start by considering each of the given directions and translating them into vectors on a grid where east is the positive x-axis and north is the positive y-axis:
- 3 miles east adds +3 to the x (east-west) coordinate.
- 6 miles southeast is equivalent to adding +4.24 to the x coordinate and -4.24 to the y (south-north) coordinate because of the 45° angle that southeast makes with the axes (6 miles / √2).
- 5 miles south subtracts 5 from the y coordinate.
- 5 miles southwest is equivalent to subtracting 3.54 from both the x and y coordinates because of the 45° angle that southwest makes with the axes (5 miles / √2).
- 2 miles east adds +2 to the x coordinate.
Adding these vectors together gives us a total displacement.
For the x-coordinate (east-west direction), the sum is 3 + 4.24 - 3.54 + 2 = 5.7 miles east.
For the y-coordinate (south-north direction), the sum is -4.24 - 5 - 3.54 = -12.78 miles south.
The person's final displacement is thus 5.7 miles east and 12.78 miles south from the starting point.