Question 49

Mathematics Trigonometric Equations Hard

Two towers face each other separated by a distance of 25 meters. As seen from the top of the first tower, the angle of depression of the second tower’s base is 60° and that of the top is 30°. The height (in meters) of the second tower is

(A) <span class="math-tex">\(\dfrac{50}{\sqrt{3}}\)</span>
(B) <span class="math-tex">\(\dfrac{25}{\sqrt{3}}\)</span>
(C) 50
(D) <span class="math-tex">\(25\sqrt{3}\)</span>
✓ Correct Answer: Option A Verified Solution

📘 Step-by-Step Solution & Detailed Explanation:

Let the first tower have height H, and the second tower height h. The distance between them is 25 m.

  • Angle of depression 60° to the base of the second tower: \(\tan 60° = \frac{H}{25}\), so \(H = 25\sqrt{3}\).
  • Angle of depression 30° to the top of the second tower: \(\tan 30° = \frac{H - h}{25}\), so \(H - h = \frac{25}{\sqrt{3}}\).

\(h = 25\sqrt{3} - \frac{25}{\sqrt{3}} = \frac{75 - 25}{\sqrt{3}} = \frac{50}{\sqrt{3}}\) m

Answer: (A)

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