--- title: "Gradient between P and Q (constants 100, 0)" subject: "Surveying II" chapter: 2 key: "semester_4/surveying_ii:tacheometer-gradient-pq-from-r-100" source: https://www.bctengineer.com/q/semester_4/surveying_ii/tacheometer-gradient-pq-from-r-100 site: BCT Engineering --- # Gradient between P and Q (constants 100, 0) Surveying II · Chapter 2: Tacheometry - Asked in: 2070 Bhadra, 8 marks - Video: [Calculate the horizontal distance between stations and RL of B when inst. constant is 100](https://www.youtube.com/watch?v=5BOguNaD3ZA) — SBK Concept - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_4/surveying_ii/tacheometer-gradient-pq-from-r-100 - Whole chapter: https://www.bctengineer.com/md/semester_4/surveying_ii/chapter-2 ## Question The following observation were made with a tacheometer. The staff was held vertical (constants are 100 and 0). | Inst. St | Staff St | Bearing | Vertical angle | Staff Readings | |---|---|---|---|---| | R | P | 100° | +8°20' | 2.60, 1.85, 1.10 | | R | Q | 200° | -2°30' | 2.50, 1.91, 1.32 | Find the gradient between P and Q. ## Short answer Using D=Ks cos²θ and V=½Ks sin2θ, then cosine rule for PQ. PQ = 203.57 m, ΔH = -26.712 m, gradient falling 1 in 7.6 (13.12%). ## Answer $K=100$, $C=0$. The height of instrument is not given and cancels in the difference of levels. The middle reading is the axial hair reading $h$. Staff held vertical, line of sight inclined at angle $\theta$ (additive constant zero, which is the case for an anallatic lens or $C=0$): $$ D=Ks\cos^2\theta,\qquad V=\tfrac12Ks\sin2\theta $$ where $s$ is the staff intercept, $\theta$ the vertical angle and $K=100$, $C=0$. #### Step 1: Horizontal distance and vertical component of each sight **Point P** (bearing 100°00', angle of elevation $8^\circ 20'$): $$ \begin{aligned} s&=2.600-1.100=1.500\ \text{m}\\ D&=100\times1.500\times\cos^2 8^\circ 20'=146.849\ \text{m}\\ V&=\tfrac12\times100\times1.500\times\sin 2(8^\circ 20')=21.510\ \text{m} \end{aligned} $$ **Point Q** (bearing 200°00', angle of depression $2^\circ 30'$): $$ \begin{aligned} s&=2.500-1.320=1.180\ \text{m}\\ D&=100\times1.180\times\cos^2 2^\circ 30'=117.775\ \text{m}\\ V&=\tfrac12\times100\times1.180\times\sin 2(2^\circ 30')=-5.142\ \text{m} \end{aligned} $$ #### Step 2: Difference in level The height of instrument is the same for both sights, so it cancels (RL of point $=$ RL of axis $+V-h$): $$ \begin{aligned} \Delta H&=(V_{Q}-h_{Q})-(V_{P}-h_{P})\\ &=(-5.142-1.910)-(+21.510-1.850)=-26.712\ \text{m} \end{aligned} $$ #### Step 3: Horizontal distance PQ The angle at R between the two lines is the difference of the bearings, $100^\circ 00'$ (100°00' and 200°00'). $$ \begin{aligned} PQ&=\sqrt{D_1^2+D_2^2-2D_1D_2\cos\gamma}\\ &=\sqrt{146.849^2+117.775^2-2\times146.849\times117.775\cos 100^\circ 00'}=203.574\ \text{m} \end{aligned} $$ #### Step 4: Gradient $$ \text{Gradient}=\frac{\Delta H}{PQ}=\frac{26.712}{203.574}=0.13122 $$ The line from P to Q is falling: 1 in 7.6, i.e. 13.12 %, an angle of $7^\circ 28'32''$ with the horizontal. **Answer:** PQ = 203.57 m; difference in level = -26.712 m; gradient falling 1 in 7.6 (13.12 %, 7°29'). ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. The readings are 2.60, 1.85, 1.10. What is the intercept s and axial reading h? A. s = 1.500 m, h = 1.85 m B. s = 1.85 m, h = 1.500 m C. s = 3.70 m, h = 1.10 m D. s = 0.75 m, h = 2.60 m Answer: A. s = top - bottom and the middle hair is the axial reading. (Key point: Stadia hairs order) 2. Why is V negative for Q? A. The staff reading is smaller B. Q is sighted at an angle of depression of 2°30' C. The bearing exceeds 180° D. The intercept is smaller than at P Answer: B. V = (1/2)Ks sin2θ takes the sign of θ. (Key point: Sign of V) 3. What angle is used in the cosine rule? A. 300° B. 80° C. 100° (200° - 100°) D. 200° Answer: C. The included angle at R is the bearing difference. (Key point: Angle between PQ lines) 4. What is the direction and gradient of PQ? A. Rising, about 1 in 7.6 B. Rising, about 1 in 131 C. Falling, about 1 in 39 D. Falling, about 1 in 7.6 Answer: D. ΔH = -26.712 m over 203.57 m gives 13.12 % falling. (Key point: Gradient result) ## Related in Surveying II - [Subtense bar and tacheometer gradient A-B](https://www.bctengineer.com/question.md?key=semester_4%2Fsurveying_ii%3Asubtense-bar-3m-level-difference-gradient) — Surveying II, chapter 2 - [Anallatic tacheometer distance XY, gradient](https://www.bctengineer.com/question.md?key=semester_4%2Fsurveying_ii%3Aanallatic-xy-gradient-p) — Surveying II, chapter 2, asked 3× - [Anallatic tacheometer AB gradient from P](https://www.bctengineer.com/question.md?key=semester_4%2Fsurveying_ii%3Aanallatic-ab-gradient-percent-angle) — Surveying II, chapter 2, asked 3× - [Anallatic tacheometer distance XY, gradient](https://www.bctengineer.com/question.md?key=semester_4%2Fsurveying_ii%3Aanallatic-xy-gradient-p-2081c) — Surveying II, chapter 2, asked 3× - [Gradient P to Q, constants 100 and 0](https://www.bctengineer.com/question.md?key=semester_4%2Fsurveying_ii%3Agradient-pq-tacheometric-constants-100-0) — Surveying II, chapter 2, asked 2× - [Anallatic tacheometer A to B distance, gradient](https://www.bctengineer.com/question.md?key=semester_4%2Fsurveying_ii%3Aanallatic-bm-b-distance-gradient-970) — Surveying II, chapter 2 ## Related in other subjects - [AVL tree for 10, 20, 30, 35, 50, ...](https://www.bctengineer.com/question.md?key=semester_4%2Fdata_structure_and_algorithm%3Aavl-build-10-20-30-35) — Data Structure and Algorithm, chapter 6 - [Delta load 20∠60° power and wattmeter readings](https://www.bctengineer.com/question.md?key=semester_1%2Fbasic_electrical_engineering%3Adelta-load-20-60-440v-two-wattmeter) — Basic Electrical Engineering, chapter 8 - [AVL tree for 10 20 30 50 45 40 8 5 3](https://www.bctengineer.com/question.md?key=semester_4%2Fdata_structure_and_algorithm%3Aavl-build-10-20-30) — Data Structure and Algorithm, chapter 6 - [Level book, staff readings at B and C (1251.250)](https://www.bctengineer.com/question.md?key=semester_3%2Fsurveying_i%3Alevel-book-15m-b-c-rl-1250-800-1251-250) — Surveying I, chapter 5, asked 3×