Trigonometry & Inverse Trigonometry
Heights and Distances
Grade 11
Question:
<p>A person standing on the bank of a river observes that the angle of elevation of the top of a tree on the opposite bank of the river is 60° and when he retires 40 metre away from the tree the angle of elevation becomes 30°. The breadth of the river is:</p>
<p>20 m</p>
<p>30 m</p>
<p>40 m</p>
<p>60 m</p>
Step-by-Step Solution
Key Concept: Set up two right triangles from the same tree with different observer positions, then use the tangent ratios to create a system of equations relating the river breadth and tree height.
<p><strong>Step 1:</strong> Let h = height of tree, d = breadth of river (distance from initial position to tree base).</p><p><strong>Step 2:</strong> From initial position: tan(60°) = h/d → h = d√3</p><p><strong>Step 3:</strong> After retreating 40 m: tan(30°) = h/(d+40) → h = (d+40)/√3</p><p><strong>Step 4:</strong> Equate the two expressions for h: d√3 = (d+40)/√3</p><p><strong>Step 5:</strong> Multiply both sides by √3: 3d = d + 40</p><p><strong>Step 6:</strong> Simplify: 2d = 40 → d = 20 metres</p><p><strong>Verification:</strong> h = 20√3; Check: tan(30°) = 20√3/(20+40) = 20√3/60 = √3/3 ✓</p><p>∴ Answer: <strong>20 metres</strong></p>
Correct Answer: A