Multiple choice

A person walking along a straight road towards a hill observes at two points distance $\sqrt{3}$ km, the angle of elevation of the hill to be $30^{0}$ and $60^{0}$. The height of the hill is

  1. $\dfrac{3}{2}$ km
  2. $\sqrt{\dfrac{2{1}}{3{2}}}$
  3. $\displaystyle \frac{(\sqrt{3}+1)}{2}$ km
  4. $\sqrt{3}$ km
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A Correct answer
Explanation

Let the distance from the nearer observation point to the hill be x km. Then h = x tan 60° and h = (x + sqrt(3)) tan 30°, which gives x = sqrt(3)/2 and h = 3/2 km.

AI explanation

Using the tangent ratio with distance x from the second point to the hill's base, we have tan 60 = h / x, so x = h / sqrt(3). For the first point, tan 30 = h / (sqrt(3) + x), giving sqrt(3) + x = h * sqrt(3). Substituting x yields sqrt(3) + (h / sqrt(3)) = h * sqrt(3), which simplifies to 3 = 2h, making the height h = 3/2 km.

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