<p>Given the angle of elevation of a cloud from a point P which is 25 m above a lake is \(30°\) and the angle of depression of the reflection of the cloud in the lake from P is \(60°\). Find the height of the cloud from the surface (in metres).</p>
Step-by-Step Solution
Key Concept: The reflection of the cloud in the lake is equidistant below the lake surface as the cloud is above it. Use the two right triangles formed by the angle of elevation and depression to set up equations involving the cloud's height and horizontal distance.
<p><strong>Step 1:</strong> Set up the problem. Let h = height of cloud above lake surface, and d = horizontal distance from P to the point directly below the cloud.</p><p><strong>Step 2:</strong> For the angle of elevation (30°) from point P (25 m above lake) to the cloud: tan(30°) = (h - 25)/d, so 1/√3 = (h - 25)/d, giving d = √3(h - 25)</p><p><strong>Step 3:</strong> For the angle of depression (60°) from point P to the reflection of the cloud: The reflection is at depth h below the lake surface. tan(60°) = (25 + h)/d, so √3 = (25 + h)/d, giving d = (25 + h)/√3</p><p><strong>Step 4:</strong> Equate the two expressions for d: √3(h - 25) = (25 + h)/√3</p><p><strong>Step 5:</strong> Multiply both sides by √3: 3(h - 25) = 25 + h</p><p><strong>Step 6:</strong> Expand and solve: 3h - 75 = 25 + h → 2h = 100 → h = 50</p><p>∴ Answer: <strong>50 metres</strong></p>
Correct Answer: 50