--- title: "RL of hill top P (A to B 112 m, RL 1290)" subject: "Surveying II" chapter: 3 key: "semester_4/surveying_ii:hill-top-rl-2077c-ab-112" source: https://www.bctengineer.com/q/semester_4/surveying_ii/hill-top-rl-2077c-ab-112 site: BCT Engineering --- # RL of hill top P (A to B 112 m, RL 1290) Surveying II · Chapter 3: Trigonometric Leveling - Asked in: 2077 Chaitra, 6 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-2077c-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. Stn | HI (m) | Target | Zenithal angle FL | Zenithal angle FR | Distance | Target height | |---|---|---|---|---|---|---| | A | 1.47 | P | 59°18' | 300°36' | A to B = 112.00 m | - | | B | 1.42 | P | 69°52' | 290°09' | | - | | B | 1.42 | A | 85°32' | 274°26' | | 1.75 m | If RL of B was 1290.00 m, find the RL of the top of hill station. ## Short answer Mean zenith angles give elevation angles: 30°39'00" from A, 20°08'30" from B. Axis RLs are 1291.420 m (B) and 1299.856 m (A). Distance x from B to P is 256.560 m; RL of hilltop P = 1385.519 m. ## Answer 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: $RL_{axis\,B} = 1290.000 + 1.42 = 1291.420$ m - Sight B to A: the vane (1.75 m above the foot) at A is at 1291.420 + 112$\tan$(4°27'00") = 1300.136 m, so the ground at A is 1298.386 m and $RL_{axis\,A} = 1298.386 + 1.47 = 1299.856$ 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 = 1291.420 m and of A = 1299.856 m: $$ x = \frac{(1299.856 - 1291.420) - 112\times 0.59258}{0.36677 - 0.59258} = 256.560\ \text{m} $$ $$ RL_{P} = 1291.420 + 256.560\times 0.36677 = 1385.519\ \text{m} $$ Check from A: $1299.856 + (256.560 - 112)\times 0.59258 = 1385.519$ m (same). **Answer: RL of the top of the hill P = 1385.519 m.** ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. For B to A, FL = 85°32' and FR = 274°26'. What elevation angle is used? A. 85°33' B. 4°28' C. 4°27' D. 5°27' Answer: C. 360° - 274°26' = 85°34'; mean zenith = 85°33'; elevation = 90° - 85°33' = 4°27'. (Key point: Mean elevation B to A) 2. The vane level at A is 1300.136 m (axis B + 112 tan 4°27'), the vane is 1.75 m above the foot, and HI of A = 1.47 m. What is axis A? A. 1300.136 + 1.47 = 1301.606 m B. 1300.136 + 1.75 + 1.47 = 1303.356 m C. 1300.136 - 1.47 = 1298.666 m D. 1300.136 - 1.75 + 1.47 = 1299.856 m Answer: D. Ground at A = 1300.136 - 1.75 = 1298.386 m. Axis A = ground + HI = 1299.856 m. (Key point: Axis RL of A) 3. What does the elevation from B to A tell us? A. B is higher than A, so A is the farther station B. A is higher than B, so B is the farther station and D = 112 m is taken as the horizontal AB C. A and B are at the same level D. P is below both stations Answer: B. A sight upward from B to A means A is higher. B is taken as the farther station, with A between B and P. (Key point: Meaning of the elevation sight) 4. With x = 256.560 m as the distance from B to P, what is the horizontal distance from A to P? A. 256.560 m B. 368.560 m C. 144.560 m D. 112.000 m Answer: C. A lies between B and P, so A to P = x - D = 256.560 - 112 = 144.560 m. (Key point: Meaning of x) ## Close variants set in other exams - [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 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× ## Related in Surveying II - [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, 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 tower top Q from stations P and R](https://www.bctengineer.com/question.md?key=semester_4%2Fsurveying_ii%3Atower-top-q-from-p-and) — 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× - [RL of temple top from stations A and B](https://www.bctengineer.com/question.md?key=semester_4%2Fsurveying_ii%3Atemple-top-rl-two-stations) — Surveying II, chapter 3