Kripal starts driving from his home and drives 37 km towards south. From there he turns left and drives 31 km. Then he turns right and drives 27 km. Then he turns left and drives a certain distance of Y km. After than, he turns left and drives 57 km. Now he finally turn left and drives 58 km to reach a bank, which is exactly 7 km south of his home. Find the value of Y in Km.
27
Understanding Direction and Distance Problems
Direction and distance problems are common in reasoning tests. They involve following a set of instructions about movement, including directions and distances, to find a final position relative to the starting point, or calculating the total distance covered, or finding an unknown distance.
Tracking Kripal's Journey Step-by-Step
To solve this problem, we can track Kripal's position at each step. It is helpful to use a coordinate system, assuming his home is at the origin (0,0). Let's consider East as the positive x-direction and North as the positive y-direction. Consequently, West will be the negative x-direction and South will be the negative y-direction.
| Step | Movement | Direction | Distance (km) | Change in Position (\(\Delta x\), \(\Delta y\)) | Current Position (x, y) |
|---|---|---|---|---|---|
| Start | From Home | - | - | (0, 0) | (0, 0) |
| 1 | Drives South | South | 37 | (0, \(-37\)) | (0, \(-37\)) |
| 2 | Turns Left (from South, faces East) | East | 31 | (\(+31\), 0) | (\(0 + 31\), \(-37 + 0\)) = (\(31\), \(-37\)) |
| 3 | Turns Right (from East, faces South) | South | 27 | (0, \(-27\)) | (\(31 + 0\), \(-37 - 27\)) = (\(31\), \(-64\)) |
| 4 | Turns Left (from South, faces East) | East | Y | (\(+Y\), 0) | (\(31 + Y\), \(-64\)) |
| 5 | Turns Left (from East, faces North) | North | 57 | (0, \(\(+57\)\)) | (\(31 + Y + 0\), \(-64 + 57\)) = (\(31 + Y\), \(-7\)) |
| 6 | Turns Left (from North, faces West) | West | 58 | (\(-58\), 0) | (\(31 + Y - 58\), \(-7 + 0\)) = (\(Y - 27\), \(-7\)) |
Analysing the Final Position
After completing all the turns and distances, Kripal's final position is at the coordinates (\(Y - 27\), \(-7\)) relative to his home.
The problem states that the bank he reaches is exactly 7 km South of his home.
Since Home is at (0,0), a point 7 km South of Home is at the coordinates (0, \(-7\)).
Equating Positions to Find Y
The final calculated position of Kripal must be the same as the given position of the bank. Therefore, we can equate the coordinates:
Calculated Position = Given Bank Position
(\(Y - 27\), \(-7\)) = (0, \(-7\))
To find the value of Y, we equate the corresponding x-coordinates and y-coordinates.
Equating x-coordinates:
\(Y - 27 = 0\)
Equating y-coordinates:
\(-7 = -7\)
The y-coordinate equation is consistent, confirming our calculations for vertical displacement are correct.
Solving for the Unknown Distance Y
Now we solve the equation for Y from the x-coordinates:
\(Y - 27 = 0\)
To isolate Y, add 27 to both sides of the equation:
\(Y = 0 + 27\)
\(Y = 27\)
So, the value of the unknown distance Y is 27 km.
Conclusion
By tracking Kripal's movement using coordinates and using the given information about the bank's final position, we determined that the value of Y is 27 km.
Revision Table: Key Elements in Solving Direction Problems
| Element | Importance | How Used Here |
|---|---|---|
| Starting Point | Reference for all movements. | Set as the origin (0,0). |
| Directions (N, S, E, W) | Define the axis of movement. | Used to determine \(\Delta x\) and \(\Delta y\). |
| Distances | Magnitude of movement along a direction. | Added/subtracted to update coordinates. |
| Turns (Left/Right) | Change in facing direction. | Essential to determine the direction of the next segment. |
| Coordinate System | Method to track position changes. | Used (x, y) coordinates; East=+X, South=-Y etc. |
| Final Relative Position | Given position of the destination relative to the start. | Used to set up the equation(s) to solve for unknowns. |
Additional Information: Understanding Displacement vs. Distance
In problems like this, it's important to distinguish between distance and displacement.
- Distance: The total length of the path traveled. In Kripal's journey, the total distance would be \(37 + 31 + 27 + Y + 57 + 58\) km. This is the sum of the lengths of all segments.
- Displacement: The straight-line distance and direction from the starting point to the ending point. It is a vector quantity. In this problem, the final displacement is the vector from Home (0,0) to the Bank (0, -7), which is 7 km South.
Using the coordinate system helps in finding the net displacement easily, as it represents the final position vector from the origin.
Similar Questions
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Important Questions from Direction and Distance Turns
After leaving school, Hemant and Anurag walk towards the north for 1 km to reach a bus stop. From the bus stop, Hemant turns left and walks for 500 m. Then he turns towards his left and walks for 2 km to reach his home. From the bus stop, Anurag turns right and walks for 250 m. Then he turns right and walks for 1 km to reach his home. In which direction is Anurag's home from Hemant's home?
A bank employee drives 10 km towards South from her house and turns to her left and drives another 20 km. She again turns left and drives 40 km, then she turns to her right and drives for another 5 km. She again turns to her right and drives another 30 km to reach her bank where she works. What is the shortest distance between her bank and her house ?
A woman runs 12 km towards her North, then 6 km towards her South and then 8 km towards her East. In which direction is she from her starting point ?
Two friends Xand Ystart running and they run together for 50 m in the same direction and reach a point. Xturns right and runs 60 m, while Yturns left and runs 40 m. Then Xturns left and runs 50 m and stops, while Yturns right and runs 50 m and then stops. How far are the two friends from each other now?
Ashish cycles 11 km south, then turns west and cycles 8 km, then turns north and cycles 6 km, then turns east and cycles 2 km, then turns to his left and cycles 5 km. Where is he now with reference to his starting position?
