--- title: "RL of hill top Q from stations P and R" subject: "Surveying II" chapter: 3 key: "semester_4/surveying_ii:hill-top-q-from-p-and" source: https://www.bctengineer.com/q/semester_4/surveying_ii/hill-top-q-from-p-and site: BCT Engineering --- # RL of hill top Q from stations P and R Surveying II · Chapter 3: Trigonometric Leveling - Asked in: 2076 Baisakh, 6 marks - Video: [Trigonometric Levelling: Case-II Object & Instrument lies in Same vertical plane Board Exam Solution](https://www.youtube.com/watch?v=cag1wI8Qd0o) — OPengineer - 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-q-from-p-and - Whole chapter: https://www.bctengineer.com/md/semester_4/surveying_ii/chapter-3 ## Question The top of the hill station Q was sighted from two instrument stations P and R located at very different level and in the same vertical plane to the target and following observations were noted. | Inst st'n | HI (m) | Target | Zenith Angle FL | Zenith Angle FR | Distance | Target ht. | |---|---|---|---|---|---|---| | P | 1.47 | Q | 59°18' | 300°36' | PR = 112.00 m | | | R | 1.42 | Q | 69°52' | 290°09' | | | | R | 1.42 | P | 85°32' | 274°26' | | 1.75 m (vane) | If RL of station R was 1280.00 m, find the RL of top of hill. ## Short answer Mean zenith angles reduced to elevation; axis RLs: R = 1281.420 m, P = 1289.856 m. Solving simultaneous height equations gives RL of hill top Q = 1375.519 m. ## Answer #### Mean vertical angles | Sight | Face left | 360° − FR | Mean zenith | Elevation | |---|---|---|---|---| | P to Q | 59°18' | 59°24' | 59°21'00" | 30°39'00" | | R to Q | 69°52' | 69°51' | 69°51'30" | 20°08'30" | | R to P | 85°32' | 85°34' | 85°33'00" | 4°27'00" | #### RL of instrument axes - Axis of R = RL of R + HI = 1280.000 + 1.42 = 1281.420 m - Sight R to P: elevation 4°27'. The vane (1.75 m above the foot of P) is at 1281.420 + 112 tan(4°27') = 1290.136 m, so the ground at P = 1288.386 m and axis of P = 1288.386 + 1.47 = 1289.856 m. Elevation angles to Q: from R = 20°08'30", from P = 30°39'00". Take D = 112 m as the horizontal distance RP. #### Distance and RL of Q Both stations lie in the same vertical plane as Q, with P between R and Q. Let $x$ = horizontal distance from R to Q, so the distance from P to Q is $x - D$ with $D = 112$ m. ```text Q _.--'| _.--' | P _.--' aP | R _.--' aR - - - - - - - - | |<--- D --->|<--- (x - D) --->| |<------------ x ------------>| ``` Height of Q above axis of R = $x\tan\alpha_R$ and above axis of P = $(x-D)\tan\alpha_P$. The RL of Q is the same from both stations: $$ \begin{aligned} RL_{axis\,R} + x\tan\alpha_R &= RL_{axis\,P} + (x-D)\tan\alpha_P \\ x &= \frac{(RL_{axis\,P} - RL_{axis\,R}) - D\tan\alpha_P}{\tan\alpha_R - \tan\alpha_P} \end{aligned} $$ With $\alpha_R = 20°08'30"$ ($\tan = 0.36677$), $\alpha_P = 30°39'00"$ ($\tan = 0.59258$), axis RL of R = 1281.420 m and of P = 1289.856 m: $$ x = \frac{(1289.856 - 1281.420) - 112\times 0.59258}{0.36677 - 0.59258} = 256.560\ \text{m} $$ $$ RL_{Q} = 1281.420 + 256.560\times 0.36677 = 1375.519\ \text{m} $$ Check from P: $1289.856 + (256.560 - 112)\times 0.59258 = 1375.519$ m (same). **Answer: RL of the top of the hill Q = 1375.519 m.** ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. For P to Q, face left is 59°18' and face right is 300°36'. How is the mean zenith found? A. Average of 59°18' and 300°36' B. Average of 59°18' and (360° - 300°36') = 59°24', giving 59°21' C. Difference of the two readings D. Face left only Answer: B. FR is converted by 360° - FR = 59°24'; the mean of 59°18' and 59°24' is 59°21'. (Key point: Mean zenith angle) 2. The vane at P is 1.75 m above the foot and its level is 1290.136 m. What is the axis RL of P (HI 1.47 m)? A. 1289.856 m B. 1288.386 m C. 1291.606 m D. 1290.136 m Answer: A. Ground = 1290.136 - 1.75 = 1288.386; axis = ground + 1.47 = 1289.856 m. (Key point: Axis of P) 3. With P between R and Q, which expression gives x, the horizontal distance R to Q? A. (axis P - axis R - D tan(alpha_P)) / (tan(alpha_R) - tan(alpha_P)) B. (axis R - axis P) / (tan(alpha_R) + tan(alpha_P)) C. D / (tan(alpha_R) - tan(alpha_P)) D. (axis P + axis R) / tan(alpha_R) Answer: A. Equating RL axis R + x tan(alpha_R) = RL axis P + (x - D) tan(alpha_P) and solving for x gives this. (Key point: Setting up for x) 4. What is the RL of Q, and how can it be checked? A. 1375.519 m; check from P using (x - 112) tan(alpha_P) B. 1281.420 m; check from R using the zenith angle C. 1289.856 m; no check is possible D. 1375.519 m; check by adding 1.47 m Answer: A. RL Q = 1281.420 + 256.560 × 0.36677 = 1375.519 m, and from P: 1289.856 + 144.56 × 0.59258 gives the same value. (Key point: Final RL) ## Close variants set in other exams - [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 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 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 - [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, 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 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×