The angle of elevation of a cloud from a point 200 m above a lake is $30^\circ$ and the angle of depression of its reflection in the lake is $60^\circ$. The height of the cloud is

### Concept & Reflection in Water In problems involving water reflections, the surface of the lake acts as a mirror. The depth of the reflection below the lake's surface is exactly equal to the height of the object above the surface. $$ \text{Depth of Reflection} = \text{Height of Cloud above Lake} $$ ### Step-by-Step Solution 1. Let $H$ be the absolute height of the cloud from the surface of the lake. 2. The observer is at a point $200$ m above the lake. 3. Height of the cloud above the observer's horizontal line of sight = $(H - 200)$. 4. Depth of the cloud's reflection below the lake surface = $H$. 5. Depth of the reflection below the observer's horizontal line of sight = $(H + 200)$. 6. Let the horizontal distance from the observer to the cloud be $x$. 7. Using the angle of elevation ($30^\circ$): $$ \tan(30^\circ) = \frac{H - 200}{x} \Rightarrow x = (H - 200)\sqrt{3} $$ 8. Using the angle of depression to the reflection ($60^\circ$): $$ \tan(60^\circ) = \frac{H + 200}{x} \Rightarrow x = \frac{H + 200}{\sqrt{3}} $$ 9. Equate the two expressions for $x$: $$ (H - 200)\sqrt{3} = \frac{H + 200}{\sqrt{3}} $$ 10. Multiply both sides by $\sqrt{3}$: $$ 3(H - 200) = H + 200 $$ $$ 3H - 600 = H + 200 $$ $$ 2H = 800 $$ $$ H = 400 \text{ m} $$ ### Exam Strategy & Shortcut For a generic problem where the observer height is $h$, elevation is $\alpha$, and depression is $\beta$, the cloud height $H = h \left(\frac{\tan\beta + \tan\alpha}{\tan\beta - \tan\alpha}\right)$. If the angles are specifically $30^\circ$ and $60^\circ$, this formula always simplifies to exactly $2h$. Thus, $2 \times 200 = 400$ m. ### Common Pitfall Assuming the vertical distance from the observer to the reflection is simply $H$, forgetting to add the observer's $200$ m elevation above the lake to get the full opposite side ($H + 200$). ### Final Answer Therefore, the correct answer is **400 m**.