Which of the following statement is CORRECT for estimating the corrected area from a map of shrunk factor 'F'?
📜 Understanding Map Shrinkage
Maps, especially older ones printed on paper, are not perfectly stable. Over time, due to changes in temperature and humidity, the paper can shrink or distort. This means that the distances measured on the map are no longer accurate.
Shrinkage Factor (F): This is a ratio that quantifies how much the map has shrunk. It's calculated by comparing a known distance on the map to its original, true distance.
F = Shrunk Length / True Length
Since the map has shrunk, this factor 'F' will always be a value less than 1 (e.g., 0.98).
🔢 Why the Correction is Proportional to the Square of F
When we correct for shrinkage, we need to account for how it affects both length and area measurements.
For Length:
To find the true length from a measurement on a shrunk map, you must divide by the shrinkage factor.
Correct Length = Measured Length / F
This shows that the corrected length is inversely proportional to F.
Key Concept: Area is Two-Dimensional
Area is calculated by multiplying two length dimensions (e.g., length × width). Since the map has shrunk in both directions, each dimension is affected by the factor 'F'.
Shrunk Area ≈ (True Length × F) × (True Width × F)
Shrunk Area ≈ (True Length × True Width) × F²
Shrunk Area ≈ True Area × F²
To reverse this and find the true, corrected area from the shrunk map, we must rearrange the formula:
Correct Area = Measured Area / F²
This final formula clearly shows that the corrected area is inversely proportional to the square of the shrinkage factor (F²).
📊 Summary of Relationships
| Quantity to Correct | Relationship with Shrinkage Factor (F) | Correction Formula |
|---|---|---|
| Length | Inversely proportional | Correct Length = Measured Length / F |
| Area | Inversely proportional to the square of F | Correct Area = Measured Area / F² |
🏗️ Practical Example
Imagine you are a surveyor using an old plan where a line that was originally 100 meters long now measures only 99 meters.
- Calculate the Shrinkage Factor (F):
- F = Shrunk Length / True Length = 99m / 100m = 0.99
- Calculate Corrected Area:
- You measure a rectangular parcel on this map as 5,000 m².
- Correct Area = Measured Area / F²
- Correct Area = 5,000 / (0.99)² = 5,000 / 0.9801 ≈ 5101.5 m²
Even a small 1% shrinkage in length results in an error of over 100 m² in the area calculation. This is why applying the correct formula is crucial for legal and engineering work.