Direction And Distance

To determine how far and in which direction the boy is from his house, we'll visualize his journey and calculate the resultant displacement.

1. The boy initially moves 6 km south.

2. He then travels 8 km west.

3. Finally, he travels 9 km further south.

This results in a total southward travel of km.

We now have a right triangle, with the legs measuring 15 km south and 8 km west.

To find the hypotenuse (direct distance from home), we use the Pythagorean theorem:

where km, km.

By substituting the values:

Thus, the boy is 17 km away from his house.

For the direction, since he traveled south and west, he is in the south-west direction from his house.

Therefore, the boy is 17 km, South West from his starting point.

To determine how far the woman is from her initial position, we need to analyze her path step by step using vector displacements.

Step 1: She first walks 40 m in the North-West direction. In terms of vector components, this movement can be split into:

• Northward: m
• Westward: m

Step 2: Next, she walks 90 m in the South-East direction. This can be divided into:

• Southward: m
• Eastward: m

Step 3: She further moves 30 m North. Adding up all north-south movements, we calculate her net north-south displacement:

=

=

= m South

For east-west movements:

=

=

= 25\sqrt{2}\) m East

Now, she's m South and m East from her starting point. Her resultant distance from the starting point is given by:

=

=

=

= m

Thus, given the options, the closest value is 40 m, which is the correct answer.

To solve the problem, we need to find the straight-line distance (displacement) of the man from his starting point after walking in different directions.

1. Understanding the Concepts:

- The man walks in three legs: north, east, and south.
- We want to find the shortest distance between his final position and the starting point.
- This shortest distance is the straight-line distance (displacement), which can be found using the Pythagorean theorem in a coordinate system.

2. Analyzing the Movement:

- First, he walks 3 km north.
- Then 4 km east.
- Finally, 3 km south.

Let’s consider his starting point as the origin (0,0).

3. Calculating the Final Position:

- After walking 3 km north, his position is (0, 3).
- After walking 4 km east, his position is (4, 3).
- After walking 3 km south, his position is (4, 0) because he moves back 3 km along the north-south axis.

4. Calculating the Displacement:

The displacement is the straight-line distance from the starting point (0, 0) to the final point (4, 0).

Using the distance formula:

Final Answer:

The man is 4 km away from his starting point.

To solve the problem, we need to track the person's movements step-by-step and calculate the straight-line distance from the starting point to his final position.

1. Understanding the Concepts:

- Directions: North, East, South, West with turns affecting orientation.
- Displacement: The straight-line distance from start to end point.
- Use of Pythagoras Theorem: To calculate the distance in a right-angled triangle formed by movements in perpendicular directions.

2. Given Movements:

- Walk 5 km north.
- Turn right (east) and walk 3 km.
- Turn right again (south) and walk 5 km.
- Turn left (east) and walk 2 km.

3. Trace the Path:

- After first move: 5 km north.
- After second move: 3 km east.
- After third move: 5 km south → cancels out the 5 km north, so now at starting north-south level.
- After fourth move: 2 km east, so total east movement = 3 + 2 = 5 km.

4. Calculate Displacement:

North-south displacement = 0 km (since north 5 km and south 5 km cancel out).
East-west displacement = 5 km east.
Distance from start = 5 km.

Final Answer:

The person is 5 km away from the starting point.