Triangulating an Inaccessible Tower
A surveyor must find the height of a tower whose base is on the far bank of a river. From point A on the near bank, the angle of elevation to the top T is 32°. The surveyor walks 40 m directly away from the tower along level ground to point B and measures the angle of elevation as 21°. The ground is horizontal throughout, and A, B, and the foot of the tower F are collinear. A colleague claims the height can be found without ever crossing the river, but worries that the two elevation angles alone might be insufficient because the horizontal distances are unknown.
1. Set up equations and determine the height of the tower.
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Let h be height and x = AF the horizontal distance from A to the foot. Then tan32° = h/x and tan21° = h/(x+40). From the first, x = h/tan32°. Substitute: tan21° = h/(h/tan32° + 40). Solve: h/tan21° = h/tan32° + 40, so h(1/tan21° − 1/tan32°) = 40. Numerically 1/tan21° ≈ 2.6051, 1/tan32° ≈ 1.6003, difference ≈ 1.0048. h ≈ 40/1.0048 ≈ 39.8 m. So height ≈ 39.8 m - the two angles plus the 40 m baseline are sufficient.
2. The colleague says 'two angles alone are not enough.' Evaluate this claim precisely.
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Two elevation angles alone (without a baseline) are indeed insufficient: scaling the whole configuration preserves both angles, so height is undetermined up to scale. The 40 m separation provides the needed length scale. So the colleague is right about angles alone but wrong that the survey is underdetermined - the baseline resolves it. This is a similarity/scale argument, not a computation.
3. If measurement error of ±1° is possible on the 21° reading, estimate the sensitivity of h.
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Recompute with 20° and 22°. At 20°: 1/tan20° ≈ 2.7475, diff ≈ 1.1472, h ≈ 34.9 m. At 22°: 1/tan22° ≈ 2.4751, diff ≈ 0.8748, h ≈ 45.7 m. So ±1° on the smaller angle moves h by roughly −5 m to +6 m. The estimate is sensitive because the baseline is short relative to the distance; a longer baseline would reduce error.