A treasure map instructs you to walk 27 m south, followed by 70 m east, then 64.5 m north, and finally 28.5 m east. Determine the magnitude of your total displacement, if you follow these instructions. Calculate the answer in meters (m) and rounded to three significant figures.
D=-27i + 70 + 64.5i + 28.5=98.5 + 37.5i
D = sqrt(98.5^2 + 37.5^2) =
35.3 steps at 21.5 degrees south of west
To determine the magnitude of your total displacement, you need to calculate the net vertical and horizontal distances traveled and then use the Pythagorean theorem.
Let's break down the distances traveled:
1. South: 27 m
2. East: 70 m
3. North: 64.5 m
4. East: 28.5 m
To calculate the net vertical distance:
The distance traveled north is 64.5 m, and the distance traveled south is 27 m. Subtracting the south distance from the north distance gives us:
64.5 m - 27 m = 37.5 m
The net vertical distance is 37.5 m.
To calculate the net horizontal distance:
The distance traveled east is 70 m, and the distance traveled east again is 28.5 m. Adding these distances gives us:
70 m + 28.5 m = 98.5 m
The net horizontal distance is 98.5 m.
Now, using the Pythagorean theorem, we can find the magnitude of the total displacement:
Displacement² = (net vertical distance)² + (net horizontal distance)²
Displacement² = 37.5 m² + 98.5 m²
Displacement² = 1406.25 m² + 9702.25 m²
Displacement² = 11108.50 m²
Taking the square root of both sides, we get:
Displacement ≈ √11108.50 m²
Displacement ≈ 105.440 m
Rounded to three significant figures, the magnitude of your total displacement is approximately 105.440 m.