Discussion
Two vertical poles are 200 m apart and the height of one is double that of the other. From the middle point of the line joining their feet, an observer finds the angular elevations of their tops to be
Answer: C
Let the poles be \( h \) and \( 2h \). The observer is at the midpoint, so each is 100 m away.
Complementary elevations \( \theta \) and \( 90^{\circ}-\theta \) have tangents whose product is 1, since \( \tan(90^{\circ}-\theta) = \cot\theta \):
\[ \frac{h}{100} \times \frac{2h}{100} = 1 \quad\Longrightarrow\quad 2h^{2} = 10000 \]
\[ h^{2} = 5000 \quad\Longrightarrow\quad h = 50\sqrt{2} = 70.71 \text{ m} \]
\[ 2h = 141.42 \text{ m} \]
So roughly 70.5 m and 141 m. Neither angle is ever needed - the complementarity alone pins the answer down.
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