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RL of tower top, base inaccessible

Short answer

Horizontal distances XP = 181.315 m, YP = 149.222 m from solving triangle XYP. RL from X = 1309.611 m, from Y = 1266.348 m; the two values disagree, so the mean RL of tower top = 1287.98 m.

Calculate the RL of top of tower P, base of which was not accessible from the two instrumentation stations X and Y. The top of the tower and the instrument stations were not in the same vertical plane. The observed zenithal angles from X to P and Y to P were 30°30' and 29°20' respectively. H.I. of instruments at X and Y were 1.57 m and 1.50 m respectively and distance between them was 200 m. The horizontal angles observed were: PXY = 45°45' XYP = 60°30' respectively. The back sights taken to BM with RL of 1000.00 m were 1.8 m and 0.8 m from X and Y respectively.

Assumption: the back sights are taken on the BM (RL 1000.00 m), so the axis RL of an instrument = 1000.00 + back sight. The zenith angles are used as given (elevation =90∘−z= 90^\circ - z).

Plan triangle XYP

∠XPY=180∘−(45∘45′+60∘30′)=73°45′\angle XPY = 180^\circ - (45^\circ45' + 60^\circ30') = 73°45'

              P
             / \
        XP  /   \ YP
           /     \
          X ----- Y
            200 m
XP=200sin⁡60∘30′sin⁡73°45′=181.315 m,YP=200sin⁡45∘45′sin⁡73°45′=149.222 mXP = \frac{200\sin 60^\circ30'}{\sin 73°45'} = 181.315\ \text{m}, \qquad YP = \frac{200\sin 45^\circ45'}{\sin 73°45'} = 149.222\ \text{m}

Axis levels

  • Axis of X = 1000.000 + 1.8 = 1001.800 m
  • Axis of Y = 1000.000 + 0.8 = 1000.800 m

RL of P

  • Elevation from X =90∘−30∘30′=59∘30′= 90^\circ - 30^\circ30' = 59^\circ30': hX=181.315tan⁡59∘30′=307.811h_X = 181.315\tan 59^\circ30' = 307.811 m, so RLP=1001.800+307.811=1309.611RL_P = 1001.800 + 307.811 = 1309.611 m
  • Elevation from Y =90∘−29∘20′=60∘40′= 90^\circ - 29^\circ20' = 60^\circ40': hY=149.222tan⁡60∘40′=265.548h_Y = 149.222\tan 60^\circ40' = 265.548 m, so RLP=1000.800+265.548=1266.348RL_P = 1000.800 + 265.548 = 1266.348 m

The two values differ by 43.263 m, which shows that the given data are not consistent, so the mean of the two is taken.

Answer: RL of the top of the tower P = 1287.98 m (mean of 1309.61 m and 1266.35 m).

✓ Answer checked by Claude Opus 5.5 (high effort)

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