Observed staff reading = 2.5 m on a staff improperly held with slope 20:1. The correct staff reading will be:
Correct Answer: B. 50/√401 m
📚 Detailed Explanation: Tilted Staff Correction — Correct Reading = 50/√401 m
Why B (50/√401 m) is correct: When a levelling staff is tilted (not held vertically), the observed staff reading is along the slant, not the true vertical. Using the slope ratio and similar triangles, the true vertical reading is calculated from the observed inclined reading.
Given: Observed staff reading = 2.5 m (along tilted staff)
Slope of staff = 20:1 (20 vertical units per 1 horizontal unit of lean)
→ triangle sides: vertical = 20, horizontal = 1
Slope of staff = 20:1 (20 vertical units per 1 horizontal unit of lean)
→ triangle sides: vertical = 20, horizontal = 1
Hypotenuse of slope triangle = √(20² + 1²) = √(400 + 1) = √401
Relationship (by similar triangles):
True vertical reading 20
─────────────────────── = ───────
Observed reading (along staff) √401
True vertical reading = 2.5 × (20 / √401) = 50 / √401 m
Numerical check:
√401 ≈ 20.025
50/20.025 ≈ 2.497 m (slightly less than 2.5 m — the tilt effect is small at 20:1)
Why Other Options Are Wrong
| Option | Expression | Why Incorrect |
|---|---|---|
| A. 2.8 m | 2.8 | Larger than observed reading — impossible; tilt always reduces the true vertical |
| B. 50/√401 | 50/√401 ≈ 2.497 m | CORRECT by similar triangles |
| C. 51/√399 | 51/√399 | Wrong numerator; uses wrong slope ratio |
| D. 20/7 | ≈ 2.857 m | Larger than observed; wrong approach |
- Tilted staff: slope 20:1 → hypotenuse = √401.
- True vertical = observed × (20 / √401) = 2.5 × 20 / √401 = 50/√401 m.
- A tilted staff always gives a reading slightly larger than the true vertical value.