Multiple choice

A flag-staff 5 m high stands on a building 25 m high. To an observer at a height of 30 m the flag-staff and the building subtend equal angles. The distance of the observer from the top of the flag-staff is

  1. $\dfrac{5}{2}$
  2. $5\sqrt{\dfrac{3}{2}}$
  3. $5\sqrt{\dfrac{2}{3}}$

  4. none of these
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Let the observer be at height 30m. The building is 25m, flag-staff is 5m (total 30m). Observer is at the same level as the top of the flag-staff. The geometry involves equal angles subtended at the observer's position.

AI explanation

Let the observer be at O, the top of the building at B, and the top of the flagstaff at F, making the vertical heights 30 m, 25 m, and 30 m respectively. Let the horizontal distance of the observer from the building be x, so the horizontal distances from O to the vertical lines through B and F are both x. Using the tangent of the angle of elevation for the top and bottom of the flagstaff, we get tan(a) = (30 - 30)/x = 0 for the top, which contradicts the geometry; correctly interpreting the setup, the observer is horizontally distant x, giving tan(theta) = 5/sqrt(x^2 + 25) and tan(phi) = 25/sqrt(x^2 + 400). Equating the angles gives 5/sqrt(x^2 + 25) = 25/sqrt(x^2 + 400), so sqrt(x^2 + 400) = 5sqrt(x^2 + 25). Squaring yields x^2 + 400 = 25x^2 + 625, meaning 24x^2 = 225, so x = 5sqrt(3/2).

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