Question

PQ is a post of given height a, and AB is a tower at some distance. If $\alpha$ and $\beta$ are the angles of elevation of B, the top of the tower, at P and Q respectively. Find the height of the tower and its distance from the post.

Answer 1

Given that PQ is post of its height 'a'.AB is an tower of its height 'h' and C be any point on AB then AB=PQ=a.
Distance between post and tower is 'd' then PA=QC=d and α,β are the angle elevation of P and Q respectively.
$tan\alpha =\frac{h}{d}=>d=hcot\alpha$ -------→(1)
$tan\beta =\frac{h-a}{d}=>d=(h-a)cot\beta$ -------→(2)
from 1 & 2 equations
h Cot α = (h-a) Cot β
a Cot β = h (Cot β - Cot α)
$h=\frac{acot\beta }{cot\beta -cot\alpha }$
∴ the height of the tower is $h=\frac{acot\beta }{cot\beta -cot\alpha }$
and distance between post and tower is $d=hcot\alpha .$