--- title: "RL of hill station P (A-B 120 m, RL 1280)" subject: "Surveying II" chapter: 3 key: "semester_4/surveying_ii:hill-top-rl-2074b-ab-120" source: https://www.bctengineer.com/q/semester_4/surveying_ii/hill-top-rl-2074b-ab-120 site: BCT Engineering --- # RL of hill station P (A-B 120 m, RL 1280) Surveying II · Chapter 3: Trigonometric Leveling - Asked in: 2074 Bhadra, 5 marks - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_4/surveying_ii/hill-top-rl-2074b-ab-120 - Whole chapter: https://www.bctengineer.com/md/semester_4/surveying_ii/chapter-3 ## Question Compute RL of a hill station 'P' from two instrument stations A and B at very different level with same line of sight to that of target from the following informations. | Inst St'n | Inst ht | Target | Zenith angle FL | Zenith angle FR | Distance & Target ht and RL | |---|---|---|---|---|---| | A | 1.42 | P | 65°18' | 304°36' | Distance between A and B = 120.00 m | | B | 1.47 | P | 69°52' | 290°00' | RL of B = 1280.00 m | | A | 1.42 | B | 102°52' | 257°16' | Target ht at B = 1.50 m | ## Short answer Mean elevation angles: 24°39'00" from A, 20°04'00" from B. Axis RL at A = 1308.763 m, at B = 1281.470 m. Horizontal distance x from B to P = 296.717 m; RL of hilltop P = 1389.857 m. ## Answer The face right reading of A to P is printed as 304°36', which would not agree with the face left reading (65°18'); it is taken as 294°36' (a misprint). #### Mean vertical angles | Station | Face left | Face right (360° − FR) | Mean zenith $z$ | Elevation $\alpha = 90^\circ - z$ | |---|---|---|---|---| | A to P | 65°18' | 65°24' | 65°21'00" | 24°39'00" | | B to P | 69°52' | 70°00' | 69°56'00" | 20°04'00" | | A to B | 102°52' | 102°44' | 102°48'00" | -12°48'00" | A to B is a depression (12°48'00" below the horizontal), so A is higher than B. The angle of elevation to P is also larger from A, so A is nearer to P and B is the farther station. #### RL of instrument axes - Axis of B = 1280.000 + 1.47 = 1281.470 m - Sight A to B: the vane at B is at (axis A) $- 120\tan(12°48'00")$, so ground B + 1.5 m = axis A $+ 120\tan(-12°48'00")$. Axis A = 1280.000 + 1.5 + 120 tan(12°48'00") = 1308.763 m (the HI at A is not needed, because the axis level of A comes directly from this sight) #### Distance and RL of P Both stations lie in the same vertical plane as P, with A between B and P. Let $x$ = horizontal distance from B to P, so the distance from A to P is $x - D$ with $D = 120$ m. ```text P _.--'| _.--' | A _.--' aA | B _.--' aB - - - - - - - - | |<--- D --->|<--- (x - D) --->| |<------------ x ------------>| ``` Height of P above axis of B = $x\tan\alpha_B$ and above axis of A = $(x-D)\tan\alpha_A$. The RL of P is the same from both stations: $$ \begin{aligned} RL_{axis\,B} + x\tan\alpha_B &= RL_{axis\,A} + (x-D)\tan\alpha_A \\ x &= \frac{(RL_{axis\,A} - RL_{axis\,B}) - D\tan\alpha_A}{\tan\alpha_B - \tan\alpha_A} \end{aligned} $$ With $\alpha_B = 20°04'00"$ ($\tan = 0.36529$), $\alpha_A = 24°39'00"$ ($\tan = 0.45889$), axis RL of B = 1281.470 m and of A = 1308.763 m: $$ x = \frac{(1308.763 - 1281.470) - 120\times 0.45889}{0.36529 - 0.45889} = 296.717\ \text{m} $$ $$ RL_{P} = 1281.470 + 296.717\times 0.36529 = 1389.857\ \text{m} $$ Check from A: $1308.763 + (296.717 - 120)\times 0.45889 = 1389.857$ m (same). **Answer: RL of the hill top P = 1389.857 m.** ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. Why is the face right reading 304°36' for A to P treated as a misprint? A. Because FR must always exceed 300° B. Because 360° - 304°36' = 55°24' disagrees with FL 65°18'; 294°36' fits C. Because FR should equal FL D. Because HI of A is 1.42 m Answer: B. FL and (360° - FR) must nearly agree. 360° - 294°36' = 65°24', which matches 65°18'. (Key point: Misprint in the FR reading) 2. The mean zenith from A to B is 102°48'. What does this show? A. An elevation of 12°48', so B is higher than A B. A depression of 12°48', so A is lower than B C. A depression of 12°48', so A is higher than B and B is the farther station D. An elevation of 77°12' Answer: C. A zenith over 90° means looking below the horizontal. So A is higher, and B is the farther station. (Key point: Depression A to B) 3. The vane level at B is 1280 + 1.5 = 1281.5 m and the sight from A is a depression of 12°48'. How is axis A found? A. Axis A = 1281.5 - 120 tan 12°48' B. Axis A = 1281.5 + 1.42 C. Axis A = 1281.5 - 1.5 D. Axis A = 1281.5 + 120 tan 12°48' = 1308.763 m, as A sits above the vane Answer: D. A is higher than B, so its axis is above the vane level by D tan(depression). (Key point: Axis of A for a depression) 4. If x = 296.717 m is the distance from B to P, what is the distance from A to P? A. 176.717 m B. 296.717 m C. 416.717 m D. 120.000 m Answer: A. A lies between B and P, so A to P = x - 120 = 176.717 m. (Key point: Meaning of x) ## Close variants set in other exams - [RL of hill station P (A-B 125 m, RL B 1280.50)](https://www.bctengineer.com/question.md?key=semester_4%2Fsurveying_ii%3Ahill-p-rod-2m-at-b) — Surveying II, chapter 3, asked 2× ## Related in Surveying II - [RL of hill top P (A to B 112 m, RL 1280)](https://www.bctengineer.com/question.md?key=semester_4%2Fsurveying_ii%3Ahill-top-rl-2073m-ab-112) — Surveying II, chapter 3, asked 4× - [RL of hill top Q from stations P and R](https://www.bctengineer.com/question.md?key=semester_4%2Fsurveying_ii%3Ahill-top-q-from-p-and) — Surveying II, chapter 3, asked 4× - [RL of hill top P (A to B 120 m, vane 2.5 m)](https://www.bctengineer.com/question.md?key=semester_4%2Fsurveying_ii%3Ahill-top-rl-2072m-vane-25) — Surveying II, chapter 3 - [RL of hill top from two stations (A-B 118 m)](https://www.bctengineer.com/question.md?key=semester_4%2Fsurveying_ii%3Ahill-top-rl-two-stations-2079j) — Surveying II, chapter 3, asked 4× - [RL of hill station P, A-B 150 m, target 3.5 m](https://www.bctengineer.com/question.md?key=semester_4%2Fsurveying_ii%3Ahill-station-p-vane-15-ab) — Surveying II, chapter 3, asked 2× - [RL of hill station P from stations R and Q](https://www.bctengineer.com/question.md?key=semester_4%2Fsurveying_ii%3Ahill-station-signal-r-q-p) — Surveying II, chapter 3 ## Related in other subjects - [Level book, 20 m interval, RL 1280 to 1279.924](https://www.bctengineer.com/question.md?key=semester_3%2Fsurveying_i%3Alevel-book-20m-interval-1280-1279-924) — Surveying I, chapter 5