GATE 2023 CE (CE1) – Question 65
Trigonometric levelling was carried out from two stations P and Q to find the reduced level (R. L.) of the top of hillock, as shown in the table. The distance between Stations P and Q is 55 m. Assume Stations P and Q, and the hillock are in the same vertical plane. The R. L. of the top of the hillock (in m) is _____ (round off to three decimal places).
| Station | Vertical angle of the top of hillock | Staff reading on benchmark | R. L. of benchmark |
|---|---|---|---|
| P | 18° 45' | 2.340 m | 100.000 m |
| Q | 12° 45' | 1.660 m |
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Show answer and explanation
Correct answer: 136.940 to 138.316
Explanation
The height of the instrument at P is $100.000 + 2.340 = 102.340$ m, and at Q it is $100.000 + 1.660 = 101.660$ m (the benchmark is read from both stations). The larger angle at P means that P is closer to the hillock: let the distance be $d$ from P and $d + 55$ from Q. Then $102.340 + d\tan 18.75° = 101.660 + (d + 55)\tan 12.75°$. With $\tan 18.75° = 0.33952$ and $\tan 12.75° = 0.22628$: $0.11324d = 12.4454 - 0.68$, so $d = 103.954$ m. The R.L. of the top is $102.340 + 103.954 \times 0.33952 = 137.628$ m.