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 complementary. Find the heights of the poles.
Answer: D
Let the poles be \( h \) and \( 2h \). The observer stands at the midpoint, so each pole is 100 m away.
Complementary angles mean the two elevations are \( \theta \) and \( 90^{\circ} - \theta \), and since \( \tan(90^{\circ}-\theta) = \cot\theta \), their tangents multiply to 1:
\[ \frac{h}{100} \times \frac{2h}{100} = 1 \]
\[ 2h^{2} = 10000 \quad\Longrightarrow\quad h^{2} = 5000 \quad\Longrightarrow\quad h = 50\sqrt{2} = 70.71 \text{ m} \]
\[ 2h = 141.42 \text{ m} \]
So the poles are about 70.5 m and 141 m. The identity \( \tan\theta\cot\theta = 1 \) is what makes the problem solvable without knowing either angle.
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