Discussion
The angle of elevation of the top of a tower from a certain point is 30°. If the observer moves 40 m towards the tower, the angle of elevation of the top of the tower increases by 15°. The height of the tower is:
Answer: D
The elevation rises from \( 30^{\circ} \) to \( 30^{\circ} + 15^{\circ} = 45^{\circ} \). Writing the two horizontal distances in terms of the height \( h \):
\[ d_{30} = \frac{h}{\tan 30^{\circ}} = h\sqrt{3}, \qquad d_{45} = \frac{h}{\tan 45^{\circ}} = h \]
The observer walked the difference, 40 m:
\[ h\sqrt{3} - h = 40 \quad\Longrightarrow\quad h(\sqrt{3}-1) = 40 \]
\[ h = \frac{40}{\sqrt{3}-1} = \frac{40(\sqrt{3}+1)}{2} = 20(\sqrt{3}+1) \]
\[ h = 20 \times 2.732 = 54.64 \approx 54.6 \text{ m} \]
"Increases by 15°" means the new angle is 45°, not 15° - reading it as the latter is the trap.
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