Understanding the Direction and Distance Problem

This problem involves tracking movement in different directions and calculating the final distance from a specific point. We need to determine Shruti's final position based on her steps and turns, and then find the distance between her final position and the pole.

Step-by-Step Movement Analysis

Let's represent Shruti's home as the starting point, which we can assume is at the origin (0, 0) on a coordinate plane. North is positive Y, South is negative Y, East is positive X, and West is negative X.

1. Starting Point: Shruti's home is at (0, 0).

2. Movement 1: Walks 47 m towards the west.

   • Starting from (0, 0).
   • Moving West means decreasing the X coordinate.
   • New position: (0 - 47, 0) = (-47, 0).

3. Movement 2: Turns right and walks 18 m.

   • From (-47, 0), turning right when facing West means turning North.
   • Moving North means increasing the Y coordinate.
   • New position: (-47, 0 + 18) = (-47, 18).

4. Movement 3: Turns left and walks 53 m.

   • From (-47, 18), turning left when facing North means turning West.
   • Moving West means decreasing the X coordinate.
   • New position: (-47 - 53, 18) = (-100, 18).

5. Movement 4: Takes a final left turn and walks 30 m.

   • From (-100, 18), turning left when facing West means turning South.
   • Moving South means decreasing the Y coordinate.
   • Final position: (-100, 18 - 30) = (-100, -12).

So, Shruti's final position is at coordinates (-100, -12).

Locating the Pole

The pole is exactly 12 m to the south of her home.

• Home is at (0, 0).
• South means decreasing the Y coordinate.
• Pole's position: (0, 0 - 12) = (0, -12).

The pole is at coordinates (0, -12).

Calculating the Distance from Shruti to the Pole

We need to find the distance between Shruti's final position (-100, -12) and the pole's position (0, -12).

We can use the distance formula between two points \((x_1, y_1)\) and \((x_2, y_2)\): \(\text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\) Let \((x_1, y_1) = (-100, -12)\) (Shruti's final position) and \((x_2, y_2) = (0, -12)\) (Pole's position).

Substitute the values into the formula:

\(\text{Distance} = \sqrt{(0 - (-100))^2 + (-12 - (-12))^2}\) \(\text{Distance} = \sqrt{(0 + 100)^2 + (-12 + 12)^2}\) \(\text{Distance} = \sqrt{(100)^2 + (0)^2}\) \(\text{Distance} = \sqrt{10000 + 0}\) \(\text{Distance} = \sqrt{10000}\) \(\text{Distance} = 100\)

The distance is 100 meters.

Final Answer Determination

Based on the calculation, Shruti is 100 meters away from the pole.

Revision Table: Key Movements and Positions

Additional Information on Direction and Distance Problems

Direction and distance problems often involve visualizing paths on a plane. Here are some key concepts:

• Cardinal Directions: North, South, East, West are the four main directions.
• Turns: A right turn from North leads to East, from East to South, from South to West, and from West to North. A left turn is the opposite. A 90° turn simplifies this, aligning movement with cardinal directions.
• Coordinate System: Using a 2D coordinate system (X-Y plane) is a common method to solve these problems. The origin (0,0) is usually the starting point.
• Displacement vs. Distance: Displacement is the shortest distance between the start and end points (a vector). Distance is the total length of the path traveled (a scalar). This problem asks for the distance between the final position and the pole, which is a straight-line distance between two points, calculated using the distance formula derived from the Pythagorean theorem.
• Pythagorean Theorem: For a right-angled triangle with sides a, b, and hypotenuse c, \(a^2 + b^2 = c^2\). The distance formula is a direct application of this theorem in a coordinate plane.