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 increases by 15°. The height of the tower is:
Answer: B
"Increases by \( 15^{\circ} \)" means the new elevation is \( 30^{\circ} + 15^{\circ} = 45^{\circ} \), not \( 15^{\circ} \).
\[ d_{30} = \frac{h}{\tan 30^{\circ}} = h\sqrt{3}, \qquad d_{45} = \frac{h}{\tan 45^{\circ}} = h \]
The 40 m walked is the difference:
\[ h\sqrt{3} - h = 40 \quad\Longrightarrow\quad h = \frac{40}{\sqrt{3}-1} = \frac{40(\sqrt{3}+1)}{2} = 20(\sqrt{3}+1) \]
\[ h = 20 \times 2.732 = 54.64 \approx 54.6 \text{ m} \]
Rationalising the denominator by multiplying top and bottom by \( \sqrt{3}+1 \) keeps the arithmetic exact until the final step.
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