Trigonometry & Inverse Trigonometry
Heights and Distances
Grade 11
Question:
<p>A man is walking towards a vertical pillar in a straight path, at a uniform speed. At a certain point <em>A</em> on the path, he observes that the angle of elevation of the top of the pillar is 30°. After walking for 10 min from <em>A</em> in the same direction, at a point <em>B</em>, he observes that the angle of elevation of the top of the pillar is 60°. Then, the time taken (in minutes) by him, from <em>B</em> to reach the pillar, is</p>
<p>5</p>
<p>6</p>
<p>10</p>
<p>20</p>
Step-by-Step Solution
Key Concept: Use the relationship between angles of elevation and distances. Since the man walks in a straight line toward the pillar, the horizontal distances at points A and B satisfy: h/distance_A = tan(30°) and h/distance_B = tan(60°), where h is the pillar height. The ratio of distances gives the ratio of times at constant speed.
<p><strong>Step 1:</strong> Let h be the height of the pillar. Let the perpendicular distance from A to the pillar be d₁, and from B to the pillar be d₂.</p><p><strong>Step 2:</strong> From angle of elevation at A: tan(30°) = h/d₁ ⟹ d₁ = h/tan(30°) = h√3</p><p><strong>Step 3:</strong> From angle of elevation at B: tan(60°) = h/d₂ ⟹ d₂ = h/tan(60°) = h/√3</p><p><strong>Step 4:</strong> Distance walked from A to B: d₁ - d₂ = h√3 - h/√3 = h(√3 - 1/√3) = h(3-1)/√3 = 2h/√3</p><p><strong>Step 5:</strong> This distance was covered in 10 minutes, so speed = (2h/√3)/10 = h/(5√3)</p><p><strong>Step 6:</strong> Distance from B to pillar = d₂ = h/√3</p><p><strong>Step 7:</strong> Time from B to pillar = (h/√3)/(h/(5√3)) = (h/√3) × (5√3/h) = 5 minutes</p><p><strong>∴ Answer: A (5 minutes)</strong></p>
Correct Answer: A