The shadow of a tower becomes x metre longer, when the angle of elevation of sun changes from 60° to ?. If the height of the tower is \(\sqrt{3}x\) metre, then which one of the following is correct ?

0 < ? < 30°

30° < ? < 45°

45° < ? < 60°

60° < ? < 90°

Solution:

Formula used:1. tan ? = Perpendicular/Base2. tan ? increases with the increase of ? when ? ? (0, 90°)tan 30° = 1/?3 ? 0.57tan 45° = 1tan 60° = ?3 ? 1.732 Calculation:In an ? ABC, tan 60° = AB/BC = ?3x/BC? ?3 = ?3x/BC? BC = x -----(1)In an ? ABDtan ? = AB/BD = ?3x/(BC + x)? tan ? = ?3x/2x? tan ? = ?3/2 ? 0.86 Since, 0.57 < 0.86 < 1? 30° < ? < 45°

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