Two vertical poles stand on level ground. From a point P on the ground the top of pole A, which is 30 m tall, is seen at an elevation of 45°, while the top of pole B, which is 45 m tall, is seen at an elevation of 30°. Which pole is nearer to P, and by how much?

Each distance comes from its own pole: P is 30·cot45° = 30 m from A and 45·cot30° = 45√3 ≈ 77.94 m from B. So A is the nearer pole, by 45√3 − 30 ≈ 47.94 m. The trap is to compare the two angles alone — a bigger elevation does mean a nearer object only when the heights match, and here they do not, so both distances must actually be computed. Comparing the heights instead of the distances gives 'by 15 m', and adding the distances rather than subtracting gives 45√3 + 30 ≈ 107.9 m.