Question

The following consecutive staff readings (in m) were taken with a dumpy level and a levelling staff at a common interval of 20 m: 0.385; 1.030; 1.925; 2.825; 3.730; 4.850; 1.045; 2.005; 3.330; 4.580. The dumpy level was shifted after taking the sixth reading. The gradient (in %) of the ground between the first and the last location of the readings is ______ (in integer).

Hint:

Reduce levels using the Height of Instrument method across the change point, then find the fall over the total distance to get the percentage gradient.

Answer 1

Correct Answer: -5

Step 1: Use the Rise and Fall method instead of the HI method, as a cross-check.
Here each reading is compared with the one before it: if the staff reading DECREASES from one point to the next, the ground has RISEN by that difference; if it INCREASES, the ground has FALLEN by that difference. The fore sight of one setup and the back sight of the next (both taken to the same change point) are not compared to each other, since they refer to the same ground point.

Step 2: Tabulate the difference between consecutive readings within setup 1 (points 1-6).
$0.385 \to 1.030$: increase of $0.645$, a FALL of $0.645$ m.
$1.030 \to 1.925$: increase of $0.895$, FALL of $0.895$ m.
$1.925 \to 2.825$: increase of $0.900$, FALL of $0.900$ m.
$2.825 \to 3.730$: increase of $0.905$, FALL of $0.905$ m.
$3.730 \to 4.850$: increase of $1.120$, FALL of $1.120$ m.
Total fall, point 1 to point 6: $0.645+0.895+0.900+0.905+1.120=4.465$ m.

Step 3: Skip across the change point (the instrument shift is not a new ground difference), then tabulate setup 2 (points 6-9).
$1.045 \to 2.005$: increase of $0.960$, FALL of $0.960$ m.
$2.005 \to 3.330$: increase of $1.325$, FALL of $1.325$ m.
$3.330 \to 4.580$: increase of $1.250$, FALL of $1.250$ m.
Total fall, point 6 to point 9: $0.960+1.325+1.250=3.535$ m.

Step 4: Add the two stretches of fall to get the total fall from the first point to the last.
Total fall $=4.465+3.535=8.000$ m, matching the HI-method result exactly and confirming the reduction is correct.

Step 5: Convert to a gradient percentage.
Distance between the first and last of the 9 points $=8\times20=160$ m.
\[ \text{Gradient} = -\frac{8.000}{160}\times100=-5\% \]
\[ \boxed{-5\%} \]