Anurag and Bhavna started driving from the same point. Anurag drove North for 11 km, then turns right and covers 13 km. Bhavna goes West and covers 15 km, then turns right and covers 11 km. How far apart are they from each other in a straight line?
28 km
Calculating Distance Between Drivers' Final Positions
The problem asks us to find the straight-line distance between Anurag and Bhavna after they drive from the same starting point in different directions and distances. We can solve this by determining their final positions relative to the starting point and then calculating the distance.
Step-by-Step Solution: Tracking Movements
Let's assume the starting point is the origin (0, 0) on a coordinate plane. North is the positive y-axis, South is the negative y-axis, East is the positive x-axis, and West is the negative x-axis.
Anurag's Movement:
- Starts at (0, 0).
- Drives North for 11 km: This moves him 11 units up on the y-axis. His position becomes (0, 11).
- Turns Right: When facing North, a right turn is towards the East.
- Covers 13 km East: This moves him 13 units right on the x-axis from his current position. His position becomes (0 + 13, 11) = (13, 11).
Anurag's final position is (13, 11).
Bhavna's Movement:
- Starts at (0, 0).
- Goes West for 15 km: This moves her 15 units left on the x-axis. Her position becomes (-15, 0).
- Turns Right: When facing West, a right turn is towards the North.
- Covers 11 km North: This moves her 11 units up on the y-axis from her current position. Her position becomes (-15, 0 + 11) = (-15, 11).
Bhavna's final position is (-15, 11).
Calculating the Straight-Line Distance
Now we need to find the distance between Anurag's final position (13, 11) and Bhavna's final position (-15, 11).
The distance formula between two points $ (x_1, y_1) $ and $ (x_2, y_2) $ is given by $ \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} $.
Here, $(x_1, y_1) = (13, 11)$ and $(x_2, y_2) = (-15, 11)$.
Distance = $ \sqrt{(-15 - 13)^2 + (11 - 11)^2} $
Distance = $ \sqrt{(-28)^2 + (0)^2} $
Distance = $ \sqrt{784 + 0} $
Distance = $ \sqrt{784} $
Distance = $ 28 $
Alternatively, notice that both Anurag and Bhavna are at the same y-coordinate (11). This means they are on a horizontal line at a height of 11 km above the starting point's horizontal level. The distance between them is simply the difference in their x-coordinates.
Anurag's x-coordinate is 13.
Bhavna's x-coordinate is -15.
Distance = $|13 - (-15)| = |13 + 15| = |28| = 28$ km.
The straight-line distance between them is 28 km.
| Person | Starting Point | Movement 1 | Position after Mvt 1 | Movement 2 | Final Position |
|---|---|---|---|---|---|
| Anurag | (0, 0) | North 11 km | (0, 11) | Right 13 km (East) | (13, 11) |
| Bhavna | (0, 0) | West 15 km | (-15, 0) | Right 11 km (North) | (-15, 11) |
Revision Table: Key Concepts
| Concept | Description | Application in this Problem |
|---|---|---|
| Coordinate Geometry | Using (x, y) coordinates to represent positions. | Starting point (0,0), final positions (13,11) and (-15,11). |
| Directions (N, S, E, W) | Mapping directions to coordinate axes. | North = +y, South = -y, East = +x, West = -x. |
| Turns (Left/Right) | Changing direction based on current orientation. | Right turn from North is East; Right turn from West is North. |
| Straight-Line Distance | Shortest distance between two points. | Calculated using the distance formula or by observing the positions. |
Additional Information: Understanding Directions and Turns
In direction-based problems, it's crucial to visualize the movements correctly. Imagine yourself facing the direction mentioned and then making the turn.
- Facing North, a right turn is East.
- Facing South, a right turn is West.
- Facing East, a right turn is South.
- Facing West, a right turn is North.
Similarly for left turns:
- Facing North, a left turn is West.
- Facing South, a left turn is East.
- Facing East, a left turn is North.
- Facing West, a left turn is South.
Using a coordinate system helps simplify these problems, allowing you to represent each movement as a change in the x or y coordinate.
Similar Questions
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Important Questions from Direction and Distance Turns
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