The angle of elevation of a cloud from a point 60 m above a lake is 30°, and the angle of depression of its reflection is 60°. Find the height of the cloud.
120 m
Let P be the observation point at 60 m above the lake, let the cloud be at height H metres above the lake, and let d be the horizontal distance from P to the vertical line through the cloud.
From P, angle of elevation of the cloud gives \(\tan 30° = \frac{H - 60}{d}\), so \(d = (H - 60)\sqrt{3}\).
The reflection lies H metres below the lake, i.e. \(H + 60\) metres below P, giving \(\tan 60° = \frac{H + 60}{d}\), so \(d = \frac{H + 60}{\sqrt{3}}\).
Equate the two expressions for d: \((H - 60)\sqrt{3} = \frac{H + 60}{\sqrt{3}}\), giving \(3(H - 60) = H + 60\).
Simplify: \(3H - 180 = H + 60 \Rightarrow 2H = 240 \Rightarrow H = 120\).
Hence, the height of the cloud above the lake is 120 m.
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