- 2078 Bhadra · 5 marks
Area from perpendicular offsets to a hedge
Short answer
Applying the trapezoidal strip-by-strip method to the unequal intervals gives 206.49 m². The coordinate method, treating connections as straight lines, confirms 206.49 m².
The following perpendicular offsets were taken from a chain line to a hedge.
| Chainage (m) | 0 | 5.5 | 12.7 | 25.5 | 40.5 |
|---|---|---|---|---|---|
| Offset (m) | 5.25 | 6.5 | 4.7 | 5.2 | 4.2 |
Find the area enclosed by the boundary, using any two methods.
The offsets are at unequal intervals (5.5, 7.2, 12.8 and 15.0 m), so Simpson's rule cannot be used directly. Two methods that work for unequal intervals are the trapezoidal (strip) method and the coordinate method.
Method 1: Trapezoidal rule (strip by strip)
Each strip between two successive offsets is a trapezium with area .
| Strip | Width (m) | (m) | (m) | Area (m²) |
|---|---|---|---|---|
| 1 | 5.50 | 5.25 | 6.50 | 32.3125 |
| 2 | 7.20 | 6.50 | 4.70 | 40.3200 |
| 3 | 12.80 | 4.70 | 5.20 | 63.3600 |
| 4 | 15.00 | 5.20 | 4.20 | 70.5000 |
| Total | 206.4925 |
Area by trapezoidal rule = 206.493 m².
Method 2: Coordinate method
Chain line is the x-axis; the points in order round the figure are:
| Point | (m) | (m) |
|---|---|---|
| Chain start | 0.00 | 0.00 |
| Chain end | 40.50 | 0.00 |
| offset 5 | 40.50 | 4.20 |
| offset 4 | 25.50 | 5.20 |
| offset 3 | 12.70 | 4.70 |
| offset 2 | 5.50 | 6.50 |
| offset 1 | 0.00 | 5.25 |
Area = 206.493 m², which agrees with the first method.
Answer: area enclosed between the chain line, the hedge and the first and last offsets = 206.49 m² (by both methods).
✓ Answer checked by Claude Opus 5.5 (high effort)
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