Multiple choice

The angles of elevation of a cliff at a point $A$ on the ground and a point $B$, $100\ m$ vertically above $A$ are $\alpha$ and $\beta$ respectively. The height of the cliff is

  1. $\dfrac{100\, cot\, \alpha}{cot\,\alpha + \cot\,\beta}$
  2. $\dfrac{100\, cot\, \beta}{cot\,\alpha - \cot\,\beta}$
  3. $\dfrac{100\, cot\, \beta}{cot\,\beta - \cot\,\alpha}$
  4. $\dfrac{100\, cot\, \beta}{cot\,\beta + \cot\,\alpha}$
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C Correct answer
Explanation

Let height of cliff be h. From point A, cot(alpha) = distance/h. From point B, cot(beta) = distance/(h-100). Solving for h gives h = 100 * cot(beta) / (cot(beta) - cot(alpha)).

AI explanation

Let the cliff's total height from the ground be H and the horizontal distance from point A to the cliff base be x. Using trigonometric ratios, cot(alpha) = x/H and cot(beta) = x/(H - 100). Substituting x = H cot(alpha) into the second equation gives H cot(alpha) = (H - 100) cot(beta). Solving this linear equation for H yields H = 100 cot(beta) / (cot(beta) - cot(alpha)).

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