--- title: "RL of radio tower top from hill tops A, B" subject: "Surveying II" chapter: 3 key: "semester_4/surveying_ii:radio-tower-top-rl-from-hill" source: https://www.bctengineer.com/q/semester_4/surveying_ii/radio-tower-top-rl-from-hill site: BCT Engineering --- # RL of radio tower top from hill tops A, B Surveying II · Chapter 3: Trigonometric Leveling - Asked in: 2079 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/radio-tower-top-rl-from-hill - Whole chapter: https://www.bctengineer.com/md/semester_4/surveying_ii/chapter-3 ## Question The top (T) of a radio transmission tower was sighted from two small hill tops A and B which were in the line with T. Calculate the R.L. of the top of the tower and the horizontal distance from A to the tower from the following data. | Instrument Station | Height of Instrument (m) | Sighted to | Vertical Angle | Remarks | |---|---|---|---|---| | A | 1.40 | T | 26°21' | RL of B = 1121.500 m | | B | 1.500 | T | 21°18' | Distance AB = 75 m | | B | 1.500 | Target at A | 14°56' | Target ht at A = 3 m | ## Short answer Axis RLs: B = 1123.000 m, A = 1141.403 m. From elevation angles 21°18′ and 26°21′, distance B–T = 177.800 m, RL of tower top = 1192.321 m, distance A–T = 102.800 m. ## Answer A, B and T are in one vertical plane. B sees A at an angle of elevation, so A is the higher station and nearer the tower. Distance AB = 75 m is taken as horizontal. All the angles are angles of elevation. #### RL of instrument axes - Axis of B = RL of B + HI = 1121.500 + 1.5 = 1123.000 m - Sight B to A: elevation 14°56'. The vane (3 m above the foot of A) is at 1123.000 + 75 tan(14°56') = 1143.003 m, so the ground at A = 1140.003 m and axis of A = 1140.003 + 1.4 = 1141.403 m. Elevation angles to T: from B = 21°18'00", from A = 26°21'00". Take D = 75 m as the horizontal distance BA. #### Distance and RL of T Both stations lie in the same vertical plane as T, with A between B and T. Let $x$ = horizontal distance from B to T, so the distance from A to T is $x - D$ with $D = 75$ m. ```text T _.--'| _.--' | A _.--' aA | B _.--' aB - - - - - - - - | |<--- D --->|<--- (x - D) --->| |<------------ x ------------>| ``` Height of T above axis of B = $x\tan\alpha_B$ and above axis of A = $(x-D)\tan\alpha_A$. The RL of T 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 $\alpha_B = 21°18'00"$ ($\tan = 0.38988$), $\alpha_A = 26°21'00"$ ($\tan = 0.49532$), axis RL of B = 1123.000 m and of A = 1141.403 m: $$ x = \frac{(1141.403 - 1123.000) - 75\times 0.49532}{0.38988 - 0.49532} = 177.800\ \text{m} $$ $$ RL_{T} = 1123.000 + 177.800\times 0.38988 = 1192.321\ \text{m} $$ Check from A: $1141.403 + (177.800 - 75)\times 0.49532 = 1192.321$ m (same). Horizontal distance from A to the tower $= x - D = 177.800 - 75 = 102.800$ m (and from B: 177.800 m). **Answer: RL of the top of the tower T = 1192.321 m; horizontal distance from A to the tower = 102.800 m.** ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. B sights A at an angle of elevation of 14°56'. What does this tell us? A. B is higher than A B. A is higher than B and nearer the tower C. A and B are at the same level D. A is farther from the tower than B Answer: B. Looking up from B to A means A is higher; since A, B and T are in line, A lies between B and T. (Key point: Which station is higher) 2. The vane at A is 3 m above the foot and at RL 1143.003 m. Axis of A (HI 1.4 m) is? A. 1141.403 m B. 1140.003 m C. 1144.403 m D. 1143.003 m Answer: A. Ground at A = 1143.003 - 3 = 1140.003; axis = ground + 1.4 = 1141.403 m. (Key point: Axis of A) 3. If x = 177.800 m is the horizontal distance from B to T and AB = 75 m, what is the horizontal distance from A to the tower? A. 177.800 m B. 252.800 m C. 102.800 m D. 75.000 m Answer: C. A is between B and T, so AT = x - AB = 177.800 - 75 = 102.800 m. (Key point: Distance from A to tower) 4. How is the RL of T found once x is known? A. RL axis B + x tan(21°18') B. RL axis A + x tan(26°21') C. RL of B + x tan(26°21') D. RL axis B + (x - 75) tan(21°18') Answer: A. From B the height is x tan(alpha_B) above its axis: 1123.000 + 177.800 × 0.38988 = 1192.321 m. 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