Multiple choice

The top of a hill observed from the top and bottom of a building of height $h$ is at angles of elevation $\alpha$ and $\beta$ respectively. The hehill is:ight of the

  1. $\cfrac { h\cot { \beta } }{ \cot { \beta } -\cot { \alpha } } $
  2. $\cfrac { h\cot { \alpha } }{ \cot { \alpha } -\cot { \beta } } $
  3. $\cfrac { h\tan { \alpha } }{ \tan { \alpha } -\tan { \beta } } $

  4. None of the above
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B Correct answer
Explanation

Using standard trigonometric relations for the height of a hill observed from the top and bottom of a building, the height of the hill is given by h * cot(alpha) / (cot(alpha) - cot(beta)).

AI explanation

Let the total height of the hill be H and its horizontal distance from the building be x. From the bottom of the building, tan(beta) equals (H minus h) divided by x; from the top, tan(alpha) equals H divided by x. Rewriting these using cotangent gives x times cot(beta) equals H minus h, and x times cot(alpha) equals H. Substituting x equals H divided by cot(alpha) into the first equation and solving for H yields H equals h cot(alpha) divided by (cot(alpha) minus cot(beta)).

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