--- title: "RL of hill top P (A to B 112 m, RL 1280)" subject: "Surveying II" chapter: 3 key: "semester_4/surveying_ii:hill-top-rl-2073m-ab-112" source: https://www.bctengineer.com/q/semester_4/surveying_ii/hill-top-rl-2073m-ab-112 site: BCT Engineering --- # RL of hill top P (A to B 112 m, RL 1280) Surveying II · Chapter 3: Trigonometric Leveling - Asked in: 2073 Magh, 8 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-2073m-ab-112 - Whole chapter: https://www.bctengineer.com/md/semester_4/surveying_ii/chapter-3 ## Question The top of the hill station 'P' was sighted from two instrument stations A and B located at very different level and in the same vertical plane to that of target and following observations were noted. | Inst st'n | HI (m) | Target | Zenith Angle FL | Zenith Angle FR | Distance | Target ht. | |---|---|---|---|---|---|---| | A | 1.47 | P | 59°18' | 300°36' | A to B = 112.00 m | - | | B | 1.42 | P | 69°52' | 290°09' | | - | | C | 1.42 | A | 85°32' | 274°26' | | 1.75 m | If RL of "B" was 1280.00 m, find the RL of top of hill station. ## Short answer Mean elevation angles: 30°39'00" from A, 20°08'30" from B. Axis RLs are 1289.856 m (A) and 1281.420 m (B). Distance x from B to P = 256.560 m; RL of hilltop P = 1375.519 m. ## Answer The observation recorded against "C" is taken as the sight from B to A (C is a misprint for B). Station A is higher than B (the sight from B to A is an elevation). The angle of elevation to P is also larger from A, so A is nearer to P and B is the farther station. Take $D = 112$ m as the horizontal distance AB. #### Mean vertical angles | Station | Face left | Face right (360° − FR) | Mean zenith $z$ | Elevation $\alpha = 90^\circ - z$ | |---|---|---|---|---| | A to P | 59°18' | 59°24' | 59°21'00" | 30°39'00" | | B to P | 69°52' | 69°51' | 69°51'30" | 20°08'30" | | B to A | 85°32' | 85°34' | 85°33'00" | 4°27'00" | #### RL of instrument axes - Axis of B: $RL_{axis\,B} = 1280.000 + 1.42 = 1281.420$ m - Sight B to A: the vane (1.75 m above the foot) at A is at 1281.420 + 112$\tan$(4°27'00") = 1290.136 m, so the ground at A is 1288.386 m and $RL_{axis\,A} = 1288.386 + 1.47 = 1289.856$ m #### 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 $D = 112$ 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 $\alpha_B = 20°08'30"$ ($\tan = 0.36677$), $\alpha_A = 30°39'00"$ ($\tan = 0.59258$), axis RL of B = 1281.420 m and of A = 1289.856 m: $$ x = \frac{(1289.856 - 1281.420) - 112\times 0.59258}{0.36677 - 0.59258} = 256.560\ \text{m} $$ $$ RL_{P} = 1281.420 + 256.560\times 0.36677 = 1375.519\ \text{m} $$ Check from A: $1289.856 + (256.560 - 112)\times 0.59258 = 1375.519$ m (same). **Answer: RL of the top of the hill P = 1375.519 m.** ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. How does the answer treat the observation recorded against 'C'? A. As the sight from B to A with the vane at 1.75 m, since C is a misprint for B B. As a third station C at the same level as A C. It is ignored D. As a sight from A to B Answer: A. Only A and B are stations, and the vane at A is sighted from B, so C is read as B. (Key point: Observation recorded against C) 2. What is the mean angle of elevation from B to P (FL 69°52', FR 290°09')? A. 20°08' B. 19°51'30'' C. 20°08'30'' D. 69°51'30'' Answer: C. 360° - 290°09' = 69°51'; mean zenith = 69°51'30''; elevation = 90° - that = 20°08'30''. (Key point: Mean elevation) 3. The vane level at A is 1290.136 m, the vane is 1.75 m above the foot and HI of A is 1.47 m. What is axis A? A. 1290.136 + 1.75 + 1.47 = 1293.356 m B. 1290.136 - 1.47 = 1288.666 m C. 1290.136 - 1.75 - 1.47 = 1286.916 m D. 1290.136 - 1.75 + 1.47 = 1289.856 m Answer: D. Subtract the vane height to get the ground (1288.386 m), then add the HI of A. (Key point: Axis RL of A) 4. With x = 256.560 m from B to P, which expression gives the RL of P? A. 1289.856 + 256.560 tan 30°39' = 1441.90 m B. 1281.420 + 256.560 tan 20°08'30'' = 1375.519 m C. 1280.000 + 256.560 tan 20°08'30'' = 1374.10 m D. 1281.420 + 144.560 tan 20°08'30'' = 1334.44 m Answer: B. From B the horizontal distance is x and the axis is 1281.420 m. Using x from A would be wrong, since A to P is x - 112. (Key point: RL of P) ## Close variants set in other exams - [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 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 top P (A to B 112 m, RL 1290)](https://www.bctengineer.com/question.md?key=semester_4%2Fsurveying_ii%3Ahill-top-rl-2077c-ab-112) — Surveying II, chapter 3, asked 4× ## Related in Surveying II - [RL of hill station P (A-B 120 m, RL 1280)](https://www.bctengineer.com/question.md?key=semester_4%2Fsurveying_ii%3Ahill-top-rl-2074b-ab-120) — Surveying II, chapter 3, asked 2× - [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× - [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 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 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 - [RL of hill station P, A-B 150 m, RL B 1500](https://www.bctengineer.com/question.md?key=semester_4%2Fsurveying_ii%3Ahill-station-p-vane-175-ab) — Surveying II, chapter 3, asked 2× ## 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