Trigonometry & Inverse Trigonometry
Heights and Distances
Grade 11
Question:
<p>If 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 reflection of the cloud in the lake from P be 60°, then the height of the cloud (in meters) from the surface of the lake is __________ .</p>
Step-by-Step Solution
Key Concept: The cloud and its reflection are equidistant from the lake surface (symmetric about the lake). If cloud height is h, reflection appears at depth h below surface. Use the two angles from point P to create two equations involving the horizontal distance and heights.
<p><strong>Step 1:</strong> Set up the geometry. Let h = height of cloud from lake surface, and d = horizontal distance from P to the point directly below/above the cloud.</p><p><strong>Step 2:</strong> Point P is 25 m above the lake surface. The reflection of the cloud in the lake appears at depth h below the surface (by symmetry of reflection).</p><p><strong>Step 3:</strong> From angle of elevation 30° to the cloud:</p><p>tan(30°) = (h - 25)/d</p><p>1/√3 = (h - 25)/d</p><p>d = √3(h - 25) ... (1)</p><p><strong>Step 4:</strong> From angle of depression 60° to the reflection (which is at depth h below lake):</p><p>tan(60°) = (25 + h)/d</p><p>√3 = (25 + h)/d</p><p>d = (25 + h)/√3 ... (2)</p><p><strong>Step 5:</strong> Equate equations (1) and (2):</p><p>√3(h - 25) = (25 + h)/√3</p><p>3(h - 25) = 25 + h</p><p>3h - 75 = 25 + h</p><p>2h = 100</p><p>h = 50</p><p><strong>Verification:</strong> d = √3(50-25) = 25√3; also d = (25+50)/√3 = 75/√3 = 25√3 ✓</p><p>∴ Answer: <strong>50 meters</strong></p>
Correct Answer: 50