Multiple choice

A tower $AB$ leans towards west marking an angle $\alpha$ with the vertical. The angular elevation of $B$, the top most point of the tower is $\beta$ as observed from a point $C$ due east of $A$ at a distance $'d'$ from $A$. If the angular elevation of $B$ from a point $D$ due east of $C$ at a distance $2d$ from $C$ is $r$, then $2\tan \alpha$ can be given as

  1. $3\cot \beta - 2\cot \gamma$
  2. $3\cot \gamma - 2\cot \beta$
  3. $3\cot \beta - \cot \gamma$
  4. $\cot \beta - 3\cot \gamma$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Let the tower height be h and its westward displacement be h tan(alpha). From the two observation points, cot(beta) = d/h + tan(alpha) and cot(gamma) = 3d/h + tan(alpha). Eliminating d/h gives 2 tan(alpha) = 3 cot(beta) - cot(gamma), matching Option C.

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