Discussion
From a point P on a level ground, the angle of elevation of the top tower is 30º. If the tower is 100 m high, the distance of point P from the foot of the tower is:
Answer: D
The tower, the ground and the line of sight form a right triangle with the 30° angle at \( P \).
\[ \tan 30^{\circ} = \frac{\text{height}}{\text{distance}} \quad\Longrightarrow\quad \text{distance} = \frac{\text{height}}{\tan 30^{\circ}} \]
\[ d = \frac{100}{1/\sqrt{3}} = 100\sqrt{3} \]
\[ d = 100 \times 1.732 = 173.2 \text{ m} \]
A 30° elevation always puts the observer \( \sqrt{3} \) times the height away. Dividing by \( \sqrt{3} \) instead of multiplying gives about 58 m, which is the classic sign error.
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