5. An existing sewer at P is to be continued to Q and R on a falling gradient of 1 in 150 for plan distances of 27.12 m and 54.11 m consecutively, where the positions of $$ \mathrm{P}, \mathrm{Q} $$ and R are defined by wooden uprights. Given the following level observations, calculate the difference in level between the top of each upright and the position at which the top edge of each sight rail must be set at $$ \mathrm{P}, \mathrm{Q} $$ and R if a 2.5 m traveller is to be used. Level reading to staff on TBM on wall (RL 89.52 m$$ ) 0.39 \mathrm{~m} $$ Level reading to staff on top of upright at P 0.16 m Level reading to staff on top of upright at Q 0.35 m Level reading to staff on top of upright at R 1.17 m Level reading to staff on invert of existing sewer at P 2.84 m . All readings were taken from the same instrument position.

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To calculate the difference in level between the top of each upright and the position at which the top edge of each sight rail must be set at P, Q, and R, we will follow these steps:

1. **Calculate the Reduced Levels (RL) for P, Q, and R**:
   - The RL of the TBM (Temporary Benchmark) is given as 89.52 m.
   - The level readings to the staff on the top of the uprights at P, Q, and R are given, so we can calculate the RL for each upright.

2. **Calculate the RL for each upright**:
   - For upright P:
     $$
     \text{RL}_P = \text{RL}_{TBM} - \text{Level reading at P} = 89.52 \, \text{m} - 0.16 \, \text{m} = 89.36 \, \text{m}
     $$
   - For upright Q:
     $$
     \text{RL}_Q = \text{RL}_{TBM} - \text{Level reading at Q} = 89.52 \, \text{m} - 0.35 \, \text{m} = 89.17 \, \text{m}
     $$
   - For upright R:
     $$
     \text{RL}_R = \text{RL}_{TBM} - \text{Level reading at R} = 89.52 \, \text{m} - 1.17 \, \text{m} = 88.35 \, \text{m}
     $$

3. **Calculate the Invert Level of the Sewer at P**:
   - The invert level of the existing sewer at P is given as 2.84 m below the top of the upright at P.
   - Therefore, the invert level at P is:
     $$
     \text{Invert Level at P} = \text{RL}_P - 2.84 \, \text{m} = 89.36 \, \text{m} - 2.84 \, \text{m} = 86.52 \, \text{m}
     $$

4. **Calculate the Fall from P to Q and R**:
   - The sewer is to be continued on a falling gradient of 1 in 150.
   - The fall from P to Q (27.12 m) is:
     $$
     \text{Fall from P to Q} = \frac{27.12 \, \text{m}}{150} = 0.1808 \, \text{m}
     $$
   - The invert level at Q is:
     $$
     \text{Invert Level at Q} = \text{Invert Level at P} - \text{Fall from P to Q} = 86.52 \, \text{m} - 0.1808 \, \text{m} = 86.3392 \, \text{m}
     $$
   - The fall from Q to R (54.11 m) is:
     $$
     \text{Fall from Q to R} = \frac{54.11 \, \text{m}}{150} = 0.36073 \, \text{m}
     $$
   - The invert level at R is:
     $$
     \text{Invert Level at R} = \text{Invert Level at Q} - \text{Fall from Q to R} = 86.3392 \, \text{m} - 0.36073 \, \text{m} = 85.97847 \, \text{m}
     $$

5. **Calculate the Required Height of the Sight Rail**:
   - The height of the sight rail is set at the invert level plus the height of the traveller (2.5 m).
   - For P:
     $$
     \text{Sight Rail Height at P} = \text{Invert Level at P} + 2.5 \, \text{m} = 86.52 \, \text{m} + 2.5 \, \text{m} = 89.02 \, \text{m}
     $$
   - For Q:
     $$
     \text{Sight Rail Height at Q} = \text{Invert Level at Q} + 2.5 \, \text{m} = 86.3392 \, \text{m} + 2.5 \, \text{m} = 88.8392 \, \text{m}
     $$
   - For R:
     $$
     \text{Sight Rail Height at R} = \text{Invert Level at R} + 2.5 \, \text{m} = 85.97847 \, \text{m} + 2.5 \, \text{m} = 88.47847 \, \text{m}
     $$

6. **Calculate the Difference in Level Between the Top of Each Upright and the Sight Rail**:
   - For P:
     $$
     \text{Difference at P} = \text{Sight Rail Height at P} - \text{RL}_P = 89.02 \, \text{m} - 89.36 \, \text{m} = -0.34 \, \text{m}
     $$
   - For Q:
     $$
     \text{Difference at Q} = \text{Sight Rail Height at Q} - \text{RL}_Q = 88.8392 \, \text{m} - 89.17 \, \text{m} = -0.3308 \, \text{m}
     $$
   - For R:
     $$
     \text{Difference at R} = \text{Sight Rail Height at R} - \text{RL}_R = 88.47847 \, \text{m} - 88.35 \, \text{m} = 0.12847 \, \text{m}
     $$

### Summary of Results:
- Difference at P: -0.34 m
- Difference at Q: -0.3308 m
- Difference at R: 0.12847 m

These values indicate how much lower the sight rail needs to be set compared to the top of each upright at P and Q, while at R, the sight rail can be set slightly higher than the top of the upright.
The difference in level between the top of each upright and the sight rail at P is -0.34 m, at Q is -0.3308 m, and at R is 0.12847 m.

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