An area of a field is 250 square centimetres from a shrunk map having a shrunk factor of 0.9. Calculate the original area (square metres) on the field, if the original scale of the map is 1 cm = 50 m.
📝 Detailed Explanation: Area, Shrinkage, and Scale
This problem requires two main steps. First, we must correct the area on the map to account for the shrinkage. Second, we must convert this corrected map area into the actual area on the ground using the map's scale.
A common mistake is to apply the shrinkage factor directly to the area. However, the shrinkage factor (SF) is a linear ratio. Since area is a two-dimensional quantity (length × width), the shrinkage factor must be squared when dealing with areas.
⚙️ Step-by-Step Calculation
Step 1: Calculate the Original Area on the Map
The relationship between original and shrunk values is:
Shrunk Area = Original Area × (Shrinkage Factor)²
To find the original area, we rearrange the formula:
Original Map Area = Shrunk Map Area / (Shrinkage Factor)²
- Shrunk Map Area = 250 cm²
- Shrinkage Factor = 0.9
Original Map Area = 250 / (0.9)² = 250 / 0.81 ≈ 308.64 cm²
Step 2: Convert Original Map Area to Actual Field Area
The original scale is 1 cm = 50 m. To convert area, we must square the scale factor:
1 cm² on map = (50 m) × (50 m) = 2500 m² on field
Now, we convert the original map area to the field area:
Original Field Area = Original Map Area × (Scale Conversion Factor)
Original Field Area = 308.64 cm² × 2500 m²/cm²
Original Field Area = 771,600 m²
Conclusion
The calculated original area on the field is 771,600 m². This value corresponds most closely to option (d) 771,450. The minor difference is likely due to rounding in the question's source data.
💡 Key Formulas to Remember
- Shrunk Length = Original Length × SF
- Shrunk Area = Original Area × SF²
- Field Area = Map Area × (Scale)²