Discussion
The angle of elevation of the top of a tower from a point is 30 °. On walking 40 m towards the tower the angle changes to 45°.Find the height of the tower?
Answer: C
Let the tower be \( h \). The two observation points give:
\[ d_{30} = h\sqrt{3}, \qquad d_{45} = h \]
\[ h\sqrt{3} - h = 40 \quad\Longrightarrow\quad h(\sqrt{3}-1) = 40 \]
Rationalising:
\[ h = \frac{40}{\sqrt{3}-1} \times \frac{\sqrt{3}+1}{\sqrt{3}+1} = \frac{40(\sqrt{3}+1)}{2} = 20(\sqrt{3}+1) \text{ m} \]
Numerically about 54.6 m. Multiplying the numerator and denominator by the conjugate is what turns the surd denominator into the plain 2.
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