A scout troop is practicing its orientation skills with a map and compass. First they walk east for 1.2km. Next, they walk 45degrees west of north for 2.7km. In what direction must they walk to go directly back to their starting point? How far will they have to walk?
The vector to the final location from the starting point has components of 1.2 - 2.7 sin 45 = -0.709 km in the east direction and 2.7 sin 45 = 1.909 in the north direction. The direction they must walk back is the reverse of that vector. You can use the ratio of the two components to get the tangent of the angle.
The distance is the hypotenuse, sqrt[(0.709)^2 + (1.909)^2] = 2.04 km
thank you SO much!
To find the direction the scout troop must walk to go directly back to their starting point, we can use the concept of vector addition.
Given that they walked east for 1.2km and then 45 degrees west of north for 2.7km, we need to calculate the resultant displacement.
To do this, we can break down the vector components:
1. Eastward displacement: 1.2km towards the east.
2. Northward displacement: 2.7km * cos(45 degrees) towards the north.
Eastward displacement: 1.2km
Northward displacement: 2.7km * cos(45) = 1.91km
Westward displacement: 2.7km * sin(45) = 1.91km
To go directly back to their starting point, the eastward and westward displacements must cancel each other out. And the northward and southward displacements must also cancel each other out.
Thus, the total displacement will be zero.
Since the eastward and westward displacements are equal, the vector direction will be along the east-west axis. Therefore, they must walk in the west direction.
To find the total distance they need to walk, we can calculate the magnitude of the resultant displacement:
Resultant displacement = sqrt((eastward displacement)^2 + (northward displacement)^2)
= sqrt((1.2km)^2 + (1.91km)^2)
= sqrt(1.44km^2 + 3.6481km^2)
= sqrt(5.0881km^2)
= 2.26km (approximately)
Therefore, they need to walk in the west direction, and the total distance they have to walk is approximately 2.26 kilometers.
To determine the direction the scout troop must walk to go directly back to their starting point, we need to calculate the resultant displacement vector by combining the eastward and northward vectors.
Step 1: Convert the given direction angles to compass bearings:
- Walking east is equivalent to a bearing of 90 degrees.
- Walking 45 degrees west of north is equivalent to a bearing of 45 degrees north-west.
Step 2: Convert the compass bearings to Cartesian coordinates:
- East is along the positive x-axis, so the eastward vector can be represented as (1.2, 0).
- 45 degrees north-west can be represented as cos(45°) and sin(45°) in terms of x and y, which is approximately (0.707, 0.707).
Step 3: Add the eastward and north-westward vectors to get the resultant vector:
- Resultant vector = (1.2, 0) + (0.707, 0.707)
- Resultant vector = (1.907, 0.707)
Step 4: Convert the resultant vector back to compass bearings:
- Using trigonometry, we can find the angle with respect to the x-axis: arctan(0.707 / 1.907) ≈ 20.77 degrees.
- Therefore, the direction they must walk is approximately 20.77 degrees east of north.
Step 5: Find the magnitude (distance) of the resultant vector:
- The magnitude of the resultant vector can be calculated using the Pythagorean theorem: √((1.907)^2 + (0.707)^2) ≈ 2.04 km.
Thus, the scout troop must walk approximately 20.77 degrees east of north, and they will have to walk approximately 2.04 km to directly return to their starting point.