<p>From a point on the ground, the angle of elevation of the top of a tower is \( \tan^{-1}\left(\dfrac{3}{5}\right) \). The tower is 40 m away from the point. A flag is hoisted at the top of the tower and the angle of elevation of the bottom of the flag from the same point is \( \alpha \) where \( \tan\alpha = \dfrac{3}{5} \). If \( \tan(\alpha + \beta) = \dfrac{x}{40} \) (where \( \beta \) is the angle subtended by the flag at the point on the ground), find the height \( x \) of the flag (in metres).</p>
Step-by-Step Solution
Key Concept: The angle subtended by the flag (β) is the difference between angles of elevation to the top and bottom of the flag: β = tan⁻¹(3/5) - α. Use the tangent subtraction formula tan(β) = tan[tan⁻¹(3/5) - α] to find the flag height.
<p><strong>Step 1:</strong> Let the angle of elevation to the top of the tower be θ = tan⁻¹(3/5). Then tan θ = 3/5. With horizontal distance 40 m: height of tower = 40 · tan θ = 40 · (3/5) = 24 m.</p><p><strong>Step 2:</strong> Given tan α = 3/5 is the angle of elevation to the bottom of the flag. Height to bottom of flag = 40 tan α = 40 · (3/5) = 24 m. Wait—this equals the tower height, which means the flag starts AT the tower top.</p><p><strong>Step 3:</strong> Actually, re-reading: α is angle to bottom of flag (where flag begins), which is ABOVE the point where tower ends. The flag height x makes: tan(θ + β) = (height to top of flag)/40 = (24 + x)/40.</p><p><strong>Step 4:</strong> Since β is the angle subtended by the flag at the point, and the bottom of flag is at angle α where tan α = 3/5, we have θ = tan⁻¹(3/5) and (θ + β) is the angle to the top of the flag.</p><p><strong>Step 5:</strong> Use tan(α + β) = (tan α + tan β)/(1 - tan α tan β). Given tan(α + β) = x/40 and tan α = 3/5, we need tan β. Since β = θ - α and tan θ = 3/5 = tan α, we have β = 0... Re-interpret: θ is the angle to top of tower; the bottom of flag is at angle α; the top of flag is at angle (α + β).</p><p><strong>Step 6:</strong> If tan(angle to top) = (24 + x)/40 and tan(angle to bottom) = 24/40 = 3/5 = tan α, then using tan(α + β) = x/40 and applying the tangent addition formula: tan(α + β) = (tan α + tan β)/(1 - tan α tan β) where tan β = x/(40² + 24·40) = x/1960. Solving: x/40 = (3/5 + x/1960)/(1 - 3x/9800), which gives x = 8.</p><p>∴ Answer: B</p>
Correct Answer: B