Tower Survey: A Theodolite Height That Must Be Added Once
A survey team is measuring the height of a mobile tower standing on level ground. From point A the angle of elevation of the top of the tower is 30°. From point B, 40 m closer to the tower along the same straight line, the angle is 45°. The theodolite rests on a tripod, so its line of sight is 1.5 m above ground level at both stations. The tower's base is not accessible for direct measurement because of a boundary wall. A junior surveyor reports the tower as 'about 40 m tall' and the contractor's drawing shows a 60 m stay wire running from B to the top of the tower. Before the report is signed, the team must settle the height, check the stay wire, and explain to a trainee why the 1.5 m is added once and not twice. Use √3 = 1.732.
1. Find the horizontal distance from B to the tower and the true height of the tower above ground.
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Let h = height of the tower above the instrument's line of sight, and x = horizontal distance from B to the foot of the tower.
At B: tan 45° = h/x → h = x. At A: the horizontal distance is (x + 40), so tan 30° = h/(x + 40) → x + 40 = h√3.
Substituting x = h: h + 40 = 1.732h → 40 = 0.732h → h = 54.64 m, and x = 54.64 m.
Exact form: h = 40/(√3 − 1) = 20(√3 + 1) = 20 × 2.732 = 54.64 m.
Height above ground = 54.64 + 1.5 = 56.14 m ≈ 56.1 m. The horizontal distance from B is about 54.6 m.
2. Check the junior surveyor's 'about 40 m' figure and decide whether the contractor's 60 m stay wire is long enough.
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Height claim: if the tower were 40 m above the line of sight, the 45° reading would force the distance from B to be 40 m, so the distance from A would be 80 m. Then tan(angle at A) = 40/80 = 0.5, giving 26.6°, not 30°. Equivalently, a consistent 30°/45° pair with AB = 40 m needs h = 40/(√3 − 1) = 54.6 m. The 40 m figure is short by about 16 m and must not be signed.
Stay wire: from B the horizontal run is 54.64 m and the vertical rise to the top of the tower is 54.64 + 1.5 = 56.14 m (the wire runs to the top, measured from ground anchor to tower top). Required length = √(54.64² + 56.14²) ≈ 78.3 m. Even measuring to the top of the tower from the instrument level (54.64 m each way, an isosceles right triangle) gives 54.64√2 ≈ 77.3 m.
Either way the 60 m wire is too short by roughly 17–18 m; the contractor must supply at least 79 m to leave a fixing allowance.
3. A trainee writes: 'Height = 54.64 + 1.5 + 1.5 = 57.64 m.' Identify the error and give the correct statement.
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The 1.5 m describes where the instrument's horizontal line sits relative to the ground. Both the 30° and 45° angles are measured at that same raised line, so the whole triangle is drawn at instrument level; only one 1.5 m separates that triangle from the ground.
Adding it twice would require two separate vertical offsets, which do not exist here.
Correct statement: height above the instrument line = 54.64 m; add the single 1.5 m tripod offset → 56.14 m above ground.
A useful discipline: state at the start what the unknown h represents ('height above the line of sight'), solve entirely in that variable, and convert to ground level only in the last line. That way the offset is applied exactly once.