Discussion
A ship is sailing towards a lighthouse. The angle of elevation of the top of the lighthouse is observed to be 30°. After sailing 100m closer, the angle of elevation becomes 45°. Find the height of the lighthouse.
Answer: B
Let h be the height and x be the final distance from the lighthouse.
From the second observation: \(\tan 45° = h/x \Rightarrow 1 = h/x \Rightarrow h = x\).
From the first observation, the distance was x+100: \(\tan 30° = \frac{h}{x+100} \Rightarrow \frac{1}{\sqrt{3}} = \frac{h}{h+100}\) (since x=h).
\(h+100 = h\sqrt{3} \Rightarrow 100 = h(\sqrt{3}-1) \Rightarrow h = \frac{100}{\sqrt{3}-1} = \frac{100(\sqrt{3}+1)}{(\sqrt{3}-1)(\sqrt{3}+1)} = \frac{100(\sqrt{3}+1)}{3-1} = 50(\sqrt{3}+1)\) m.
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