A vertical tower stands at the edge of the far bank of a straight river. From a point at water level on the near bank, directly opposite, the angle of elevation of the top of the tower is 60°. From a point 12 m vertically above that spot — the top of an embankment wall — the elevation of the same tower top is 45°. The width of the river is:

Let the width be d. From water level the tower's height is d·tan60° = d√3. From 12 m up, the tower rises only d√3 − 12 above the eye, and that must equal d·tan45° = d. So d√3 − 12 = d, giving d(√3 − 1) = 12 and d = 12/(√3 − 1) = 6(√3 + 1) ≈ 16.39 m. (Then the tower is 28.39 m tall and 28.39 − 12 = 16.39 = d, as 45° demands.) Forgetting to halve when rationalising with the conjugate gives 12(√3 + 1) ≈ 32.8 m, rationalising the wrong way gives 6(√3 − 1) ≈ 4.4 m, and 12√3 ≈ 20.8 m treats the 12 m wall as the tower's height.