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(ii) In order to obtain the level difference between two points [tex]$A$[/tex] and [tex]$B$[/tex] on either side of a river, the method of reciprocal levelling was adopted. The following field observations were obtained:

At instrument position [tex]$1.5 \, \text{m}$[/tex] from [tex]$A$[/tex]:
- Reading on staff at [tex]$A = 1.396$[/tex]
- Reading on staff at [tex]$B = 1.387$[/tex]

At instrument position [tex]$2.5 \, \text{m}$[/tex] from [tex]$B$[/tex]:
- Reading on staff at [tex]$A = 1.837$[/tex]
- Reading on staff at [tex]$B = 1.842$[/tex]

If the R.L. of [tex]$A = 10.000 \, \text{m}$[/tex], what is the R.L. of [tex]$B$[/tex]?



Answer :

GinnyAnswer
To determine the Reduced Level (RL) of point B using the method of reciprocal levelling, we can follow these steps:

### Given Data:
- At instrument position 1, 5 m from A:
- Reading on staff at A ([tex]\( \text{reading}_{A1} \)[/tex]) = 1.396
- Reading on staff at B ([tex]\( \text{reading}_{B1} \)[/tex]) = 1.387

- At instrument position 2, 5 m from B:
- Reading on staff at A ([tex]\( \text{reading}_{A2} \)[/tex]) = 1.837
- Reading on staff at B ([tex]\( \text{reading}_{B2} \)[/tex]) = 1.842

- RL of A ([tex]\( \text{RL}_A \)[/tex]) = 10.000 m

### Solution:
1. Calculate the True Difference in Level ([tex]\( \Delta H \)[/tex])

The true difference in level considering reciprocal levelling is calculated using the formula:
[tex]\[ \Delta H = \frac{(\text{Reading on A from position 1} + \text{Reading on B from position 2}) - (\text{Reading on B from position 1} + \text{Reading on A from position 2})}{2} \][/tex]

Substituting the given readings, we get:
[tex]\[ \Delta H = \frac{(1.396 + 1.842) - (1.387 + 1.837)}{2} \][/tex]
[tex]\[ \Delta H = \frac{(3.238) - (3.224)}{2} \][/tex]
[tex]\[ \Delta H = \frac{0.014}{2} \][/tex]
[tex]\[ \Delta H = 0.007 \text{ m} \][/tex]

2. Calculate the RL of point B ([tex]\( \text{RL}_B \)[/tex])

The RL of point B with respect to the RL of point A can be calculated as:
[tex]\[ \text{RL}_B = \text{RL}_A + \Delta H \][/tex]

Given [tex]\(\text{RL}_A = 10.000 \text{ m}\)[/tex] and [tex]\( \Delta H = 0.007 \text{ m} \)[/tex], we have:
[tex]\[ \text{RL}_B = 10.000 + 0.007 \][/tex]
[tex]\[ \text{RL}_B = 10.007 \text{ m} \][/tex]

### Conclusion:
The Reduced Level (RL) of point B is [tex]\(10.007 \text{ m}\)[/tex].

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