Multiple choice

The angle of elevation of the top of a tower as observed from a point on the horizontal ground is $x$. If we move a distance $d$ towards the flux of the tower, the angle of elevation increases to $y$, then the height of the tower is

  1. $\cfrac { d\tan { x } \tan { y } }{ \tan { y- } \tan { x } } $
  2. $d\left( \tan { y } +\tan { x } \right) $
  3. $d\left( \tan { y } -\tan { x } \right) $
  4. $\cfrac { d\tan { x } \tan { y } }{ \tan { y } +\tan { x } } $
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A Correct answer
Explanation

Let h be height. tan(x) = h / (d + y_dist), tan(y) = h / y_dist. y_dist = h / tan(y). h / tan(x) = d + h / tan(y). d = h(1/tan(x) - 1/tan(y)) = h(tan(y) - tan(x)) / (tan(x)tan(y)). h = d * tan(x) * tan(y) / (tan(y) - tan(x)).

AI explanation

Let the height of the tower be h and the initial distance from the base be x. From the initial point, tan(x) equals h divided by x, so x equals h divided by tan(x). After moving a distance d towards the tower, the new distance is x minus d, meaning tan(y) equals h divided by (x minus d). Substituting the value of x into the second equation and rearranging the terms yields h equals (d multiplied by tan(x) multiplied by tan(y)) divided by (tan(y) minus tan(x)).

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