Discussion
If the angle of elevation of cloud from a point 200 m above a lake is 30º and the angle of depression of its reflection in the lake is 60º, then find the height of the cloud above the lake.
Answer: A
Let the cloud be \( H \) m above the lake surface, and let \( d \) be the horizontal distance from the observation point. The observer is 200 m above the lake, and the reflection sits \( H \) m below the surface.
Relative to the observer, the cloud is \( H - 200 \) up and the reflection is \( H + 200 \) down:
\[ \tan 30^{\circ} = \frac{H - 200}{d}, \qquad \tan 60^{\circ} = \frac{H + 200}{d} \]
Dividing eliminates \( d \):
\[ \frac{\tan 60^{\circ}}{\tan 30^{\circ}} = \frac{\sqrt{3}}{1/\sqrt{3}} = 3 = \frac{H+200}{H-200} \]
\[ H + 200 = 3H - 600 \quad\Longrightarrow\quad 2H = 800 \quad\Longrightarrow\quad H = 400 \text{ m} \]
The reflection is the mirror image in the water surface, so its depth below the surface equals the cloud's height above it - that is why the second numerator is \( H + 200 \) and not \( H - 200 \).
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