--- title: "Bowditch link traverse XABCY (X 1162.510 N)" subject: "Surveying II" chapter: 1 key: "semester_4/surveying_ii:bowditch-link-traverse-xabcy-1162" source: https://www.bctengineer.com/q/semester_4/surveying_ii/bowditch-link-traverse-xabcy-1162 site: BCT Engineering --- # Bowditch link traverse XABCY (X 1162.510 N) Surveying II · Chapter 1: Traversing - Asked in: 2068 Bhadra, 9 marks - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_4/surveying_ii/bowditch-link-traverse-xabcy-1162 - Whole chapter: https://www.bctengineer.com/md/semester_4/surveying_ii/chapter-1 ## Question Balance the coordinates of a link traverse XABCY using Bowditch's rule. The coordinates of stations X and Y are (1162.510 N, 775.250 E) and (1043.610 N, 1043.610 E) respectively and permissible closing error is 1:500. Other observed data are given below: | Lines | XA | AB | BC | CY | |---|---|---|---|---| | Length (m) | 120.00 | 111.50 | 132.40 | 97.60 | | Bearings | 135°00' | 119°30' | 175°00' | 77°30' | ## Short answer Compute latitudes and departures from WCBs; the closing error is 133.196 m (1:3.5), exceeding the 1:500 permissible limit, so the traverse should be rejected. The Bowditch-adjusted coordinates are A (1111.884 N, 854.808 E), B (1088.781 N, 946.933 E), C (994.648 N, 952.630 E). ## Answer Method: compute latitudes and departures from the given WCBs, find the closing error against the known coordinates of Y, check it against the permissible limit, then distribute it by Bowditch's rule and compute the adjusted coordinates. #### Step 1: Latitudes and departures $$ \text{Lat}=L\cos\theta,\qquad \text{Dep}=L\sin\theta $$ #### Step 2: Closing error Required change in coordinates from X to Y: $$ \begin{aligned} \Delta N_{true}&=1043.610-1162.510=-118.900\ \text{m}\\ \Delta E_{true}&=1043.610-775.250=268.360\ \text{m} \end{aligned} $$ Computed: $\Sigma \text{Lat}=-250.530$ m, $\Sigma \text{Dep}=288.723$ m. $$ \begin{aligned} e_N&=-250.530-(-118.900)=-131.630\ \text{m}\\ e_E&=288.723-(268.360)=20.363\ \text{m}\\ e&=\sqrt{e_N^2+e_E^2}=133.196\ \text{m} \end{aligned} $$ Perimeter (total length) $\Sigma L=461.50$ m, so the relative error is $e/\Sigma L = 1:3.5$. The permissible closing error is 1:500, i.e. at most 0.923 m, but the error here is 133.196 m (about 1:3.5). The given coordinates of Y and the field data do not agree, so in a real survey this traverse would be rejected and re-measured. The Bowditch adjustment is shown below only to complete the procedure. #### Step 3: Bowditch correction $$ C_{lat}=-e_N\frac{L}{\Sigma L},\qquad C_{dep}=-e_E\frac{L}{\Sigma L} $$ | Line | Length (m) | WCB | Latitude | Departure | Corr. lat | Corr. dep | Adj. lat | Adj. dep | |---|---|---|---|---|---|---|---|---| | XA | 120.00 | 135°00' | -84.853 | +84.853 | +34.227 | -5.295 | -50.626 | +79.558 | | AB | 111.50 | 119°30' | -54.905 | +97.045 | +31.802 | -4.920 | -23.103 | +92.125 | | BC | 132.40 | 175°00' | -131.896 | +11.539 | +37.763 | -5.842 | -94.133 | +5.697 | | CY | 97.60 | 77°30' | +21.125 | +95.286 | +27.838 | -4.307 | +48.962 | +90.980 | | **Sum** | 461.50 | | -250.530 | +288.723 | +131.630 | -20.363 | -118.900 | +268.360 | #### Step 4: Adjusted coordinates Start from X and add the adjusted latitude and departure of each line: | Station | Northing (m) | Easting (m) | |---|---|---| | X | 1162.510 | 775.250 | | A | 1111.884 | 854.808 | | B | 1088.781 | 946.933 | | C | 994.648 | 952.630 | | Y | 1043.610 | 1043.610 | Check: the computed coordinates of Y agree with the given Y (1043.610 N, 1043.610 E). **Answer:** adjusted coordinates A (1111.884 N, 854.808 E), B (1088.781 N, 946.933 E), C (994.648 N, 952.630 E); closing error 133.196 m (1:3.5). ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. In a link traverse X to Y, how is the closing error in northing found? A. Sum of latitudes alone B. Sum of computed latitudes minus (N of Y - N of X) C. N of Y plus N of X D. Sum of departures minus the easting difference Answer: B. The computed change in northing is compared with the true change from the known coordinates of X and Y: e_N = sum Lat - (N_Y - N_X). (Key point: Closing error in link traverse) 2. For this problem, e_N = -131.630 m and e_E = 20.363 m. What is the closing error e? A. About 111.3 m B. About 152.0 m C. About 133.2 m D. About 20.4 m Answer: C. e = sqrt(131.630^2 + 20.363^2) = 133.196 m. (Key point: Closing error value) 3. Total length is 461.50 m and permissible error is 1:500. What conclusion is drawn? A. The error 133 m (about 1:3.5) far exceeds the 0.923 m limit, so the data are inconsistent and the traverse would be rejected B. The traverse is accepted because Bowditch's rule fixes any error C. The traverse is accepted since e is less than 1:500 D. The permissible error is 133 m Answer: A. Allowed error = 461.5/500 = 0.923 m, much less than 133.196 m, so the survey fails the check. (Key point: Accuracy check) 4. In Bowditch's rule, the correction to the latitude of a line is: A. -e_N x (lat of line / sum|lat|) B. +e_N x (departure / sum|dep|) C. -e_N x (L / sum L) D. -e_E x (L / sum L) Answer: C. Corrections are proportional to line length and opposite in sign to the error: C_lat = -e_N L/sum L. (Key point: Bowditch correction) ## Close variants set in other exams - [Bowditch balancing of link traverse XABCY](https://www.bctengineer.com/question.md?key=semester_4%2Fsurveying_ii%3Abowditch-link-traverse-xabcy) — Surveying II, chapter 1, asked 2× ## Related in Surveying II - [Bowditch link traverse XABCY (X 1877.51 N)](https://www.bctengineer.com/question.md?key=semester_4%2Fsurveying_ii%3Abowditch-link-traverse-xabcy-1877) — Surveying II, chapter 1 - [Link traverse B to M, coordinates X and Y](https://www.bctengineer.com/question.md?key=semester_4%2Fsurveying_ii%3Alink-traverse-bowditch-x-y-between) — Surveying II, chapter 1 - [Closed/open traverse, Bowditch's rule](https://www.bctengineer.com/question.md?key=semester_4%2Fsurveying_ii%3Aclosed-open-traverse-bowditch-rule) — Surveying II, chapter 1, asked 2×