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Clear height of flood light tower (arena)

Short answer

Take zenith angles as elevations 2°15', 1°30', and 22°15'. D = 3/(tan2°15' - tan1°30') = 228.934 m. Plinth RL = 1203.995 m, top RL = 1293.660 m, so clear height = 89.66 m.

It is required to determine the height (clear) of a Flood light tower in an arena by using a transit theodolite and for this zenith angles observation taken at 5 m and 2 m height on a target vane held on the plinth level of tower were 87°45' and 88°30' respectively. From the same instrument, zenith angle observed at top of the tower was found as 67°45'. If the RL of the instrument axis was 1200.00 m, calculate the clear height (plinth to top) of the tower.

The vane is held at the plinth of the tower and sighted with the transit theodolite. Readings of 5 m and 2 m are heights of the vane above the plinth. RL of instrument axis = 1200.00 m. Let DD be the horizontal distance from the instrument to the tower.

                    * top (22°15')
                  / |
   axis ----------/-* 5 m  (2°15')
                    \* 2 m  (1°30')
                    |_ plinth
   |<------ D ----->|

Angles of elevation

  • α1=90∘−87∘45′=2∘15′\alpha_1 = 90^\circ - 87^\circ45' = 2^\circ15' (vane at 5 m)
  • α2=90∘−88∘30′=1∘30′\alpha_2 = 90^\circ - 88^\circ30' = 1^\circ30' (vane at 2 m)
  • α3=90∘−67∘45′=22∘15′\alpha_3 = 90^\circ - 67^\circ45' = 22^\circ15' (top)

Horizontal distance

The two vane positions are 3 m apart on the same vertical line:

D(tan⁡α1−tan⁡α2)=5−2D(\tan\alpha_1 - \tan\alpha_2) = 5 - 2 D=3tan⁡2∘15′−tan⁡1∘30′=30.03929−0.02619=228.934 mD = \frac{3}{\tan 2^\circ15' - \tan 1^\circ30'} = \frac{3}{0.03929 - 0.02619} = 228.934\ \text{m}

RL of plinth and top

  • Plinth: 1200.000+Dtan⁡α1−5=1200.000+8.995−5=1203.9951200.000 + D\tan\alpha_1 - 5 = 1200.000 + 8.995 - 5 = 1203.995 m
  • Top: 1200.000+Dtan⁡α3=1200.000+93.660=1293.6601200.000 + D\tan\alpha_3 = 1200.000 + 93.660 = 1293.660 m

Clear height

H=1293.660−1203.995=89.665 mH = 1293.660 - 1203.995 = 89.665\ \text{m}

Answer: clear height of the flood light tower = 89.66 m.

✓ Answer checked by Claude Opus 5.5 (high effort)

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