Question

PQ is a post of given height a, and AB is a tower at some distance. If α and β 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 be the tower of height H and PQ is a given post of height a, α and β are angles of elevation of the top of tower AB from P and Q. Let PA = x. PQ = a and BC = h.

The corresponding figure is as follows

In ΔQCB

`=> tan beta = h/x`

`=> x = h/(tan beta)

Again in ΔPAB

`=> tan alpha = (h + a)/x`

`=> tan alpha = ((h + a)tan beta)/h`

`=> h tan alpha = (h + a)tan beta`

`=> h(tan alpha - tan beta) = a tan beta`

`=> h =(a tan beta)/(tan alpha - tan beta)`

Now

`=> x = (a tan beta)/((tan alpha - tan beta) xx tan beta)`

`=> x = a/(tan alpha -  tan beta)`

`=> H = (a tan alpha)/(tan a - tan beta)`

Hence required heigtht is `(a tan alpha)/(tan alpha - tan beta)` And distance is  `a/(tan alpha -  tan beta)`