i walked from four miles in the direction of 30 celisus south of east, then from there i walked 3 miles to west. from there i walked 5 miles directly toward north. hw far away am i from home?'
A: USE PARALLEO=LOGRAM GRAPHING METHOD.
B: USE CALCULATION METHOD
To find out how far away you are from home, we can use the calculation method.
First, let's break down your movements:
1. You walked 4 miles in the direction of 30 degrees Celsius south of east (SE). This means you walked in the southeastern direction.
2. From there, you walked 3 miles to the west. This cancels out some of your eastward movement.
3. Finally, you walked 5 miles directly towards the north (N).
To calculate your overall displacement, we first need to find the components of your movement in the north-south (N-S) and east-west (E-W) directions.
For the N-S component:
- You walked 5 miles directly north (positive direction).
- There were no southward movements.
So, the N-S component equals +5 miles.
For the E-W component:
- You initially walked 4 miles southeast, which can be divided into east and south components using trigonometry.
- The angle between the eastward direction and your original direction is 30 degrees.
- The eastward component of the southeast movement can be found by multiplying the total southeast movement (4 miles) by the cosine of the angle.
- The southward component of the southeast movement can be found by multiplying the total southeast movement (4 miles) by the sine of the angle.
- To calculate the E-W component, we need to subtract the westward movement of 3 miles from the eastward component we just calculated.
Using trigonometry, we can find:
Eastward component = 4 miles * cos(30°) ≈ 3.464 miles
Southward component = 4 miles * sin(30°) ≈ 2 miles
E-W component = eastward component - westward movement = 3.464 miles - 3 miles = 0.464 miles
Now, we have the N-S component (+5 miles) and the E-W component (+0.464 miles).
To find the displacement or overall distance from home, we can use the Pythagorean theorem:
Displacement = √(N-S component^2 + E-W component^2)
Displacement = √(5^2 + 0.464^2)
Displacement ≈ √(25 + 0.215296)
Displacement ≈ √25.215296
Displacement ≈ 5.021 miles
Therefore, you are approximately 5.021 miles away from home.