Question

$PQ$ is post of given height $a$ and $AB$ is tower at some distance ; $\alpha$ 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

Let $AB=h$ be the height of the tower at a distance from the post $PQ=a$.

$AB$ subtends angles$\alpha$ and $\beta$ at$P$ $Q$ respectively. We have to determine $h$ and $x$ in terms of known quantity $\alpha ,\beta$ and $a$.

$AB=xtan\alpha$ and $BR=xtan\beta$

As, $PQ=AR=AB=BR$

=> $x(tan\alpha -tan\beta )$

So, $x=\frac{PQ}{tan\alpha -tan\beta }=\frac{a}{tan\alpha -tan\beta }$

=> $\frac{acos\alpha cos\beta }{sin(\alpha -\beta )}$

And,

$h=AB=xtan\alpha =\frac{cos\alpha cos\beta }{sin(\alpha -\beta )}.\frac{sin\alpha }{sin\beta }$

=> $\frac{asin\alpha sin\beta }{sin(\alpha -\beta )}$