The Shadow on the Parade Ground
At a ceremony the flag is raised at 08:00. A flagpole 6 m tall stands on level ground. A candidate 1.8 m tall stands 4 m from the base of the pole, and the sun casts his shadow 3 m long on the ground. A visitor asks how long the flagpole's shadow is and at what angle the sun's rays strike the ground. Separately, a soldier claims that because the flag is 'higher, it is more important', and proposes that the national flag be flown above a visiting unit's flag at all times, regardless of protocol.
1. Using similar triangles, compute the flagpole's shadow length and the sun's angle of elevation.
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The candidate and the flagpole are both vertical on level ground, so their triangles with their shadows are similar.
Height / shadow is constant: 1.8 / 3 = 0.6. Therefore the pole's shadow = 6 / 0.6 = 10 m.
Angle of elevation θ satisfies tan θ = height / shadow = 1.8 / 3 = 0.6, so θ = arctan(0.6) ≈ 31° above the horizon. (The same angle applies to the pole: 6 / 10 = 0.6.)
The 4 m distance from the pole is irrelevant to the shadow calculation - it tells you where the candidate stands, not how the triangles relate. Recognising which given numbers are needed is part of the problem.
2. Assess the soldier's claim about flag precedence.
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The claim confuses height with precedence. Precedence in flag protocol is governed by rules about position and order of honour, not by physically flying one flag higher as a display of superiority.
A national flag is honoured by its place of precedence and by the respect shown to it; deliberately flying it above another nation's or unit's flag in a context where protocol does not require it is a discourtesy, not a demonstration of importance. The correct response is to explain the distinction: respect for the national flag is shown by correct procedure, and incorrect procedure dishonours it rather than elevating it.
3. The visitor says the shadow 'proves' the flagpole is exactly three times the candidate's height plus a bit. Is the arithmetic conclusion sound?
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Partly. The ratio is exactly 6 / 1.8 = 3.33…, i.e. the pole is 10/3 times the candidate's height - which follows directly from the similar triangles and is sound given level ground and a vertical pole.
The reasoning is only as good as its assumptions. If the ground slopes, or the pole is not truly vertical, or the two shadows were measured at different times, the similarity breaks down and the ratio no longer holds. The correct habit is to state the result together with its conditions: 'on level ground, with both objects vertical and measured at the same moment, the pole is 10/3 times his height.'