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2057 Chaitra · 10 marks

Level book, RL of A 380.500, 3.845 on B

Short answer

From A (RL 380.500 m) applying the HI method forward yields RL of B = 371.270 m. The difference of 9.230 m over 360 m gives a falling gradient of 1 in 39 (about 2.56%).

The following consecutive readings were taken with a level and a 4 metre levelling staff on a continuously sloping ground at a common interval of 30 metres.

0.855 (on A), 1.545, 2.353, 3.115, 3.825, 0.455, 1.380, 2.055, 2.855, 3.455, 0.585, 1.015, 1.850, 2.755, 3.845 (on B).

The R.L. of A was 380.500. Make a level book and apply usual checks. Determine the gradient of the line AB.

Assumptions: the staff positions are 30 m apart; the readings 3.825 and 3.455 are foresights on change points, and the next readings (0.455 and 0.585) are backsights on the same points after the instrument is shifted. This gives 13 points, with A to B = 12 x 30 = 360 m. RL of A = 380.500 m is known. The third reading is printed as 2.353 and has been used as printed (in a similar paper it is 2.335).

Point	Chainage	BS	IS	FS	HI	Fall	RL (m)
A	0	0.855			381.355		380.500
2	30		1.545			0.690	379.810
3	60		2.353			0.808	379.002
4	90		3.115			0.762	378.240
5	120	0.455		3.825	377.985	0.710	377.530
6	150		1.380			0.925	376.605
7	180		2.055			0.675	375.930
8	210		2.855			0.800	375.130
9	240	0.585		3.455	375.115	0.600	374.530
10	270		1.015			0.430	374.100
11	300		1.850			0.835	373.265
12	330		2.755			0.905	372.360
B	360			3.845		1.090	371.270

HI is shown on the row where a new set-up starts. RL = previous RL + rise - fall.

Arithmetic checks

\begin{aligned} \Sigma BS - \Sigma FS &= 1.895 - 11.125 = -9.230 \text{ m}\\ \Sigma \text{Rise} - \Sigma \text{Fall} &= 0.000 - 9.230 = -9.230 \text{ m}\\ \text{Last RL} - \text{First RL} &= 371.270 - 380.500 = -9.230 \text{ m} \end{aligned}

All three are equal, so the arithmetic is correct.

Gradient of the line AB

RL of A = 380.500 m, RL of B = 371.270 m, difference = 9.230 m (falling from A to B).

\text{Gradient} = \frac{9.230}{360} = \frac{1}{39.00}

Answer: RL of B = 371.270 m; the line AB falls at 1 in 39.0 (2.56 %).

✓ Answer checked by Claude Opus 5.5 (high effort)

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Level book, RL of A 380.500, 3.845 on B

Arithmetic checks

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