Question

A vertical tower stand on horizontel plane and is surmounted by a vertical flagstaff of height h metre. At a point on the plane, the angle of elevation of the bottom of the flagstaff is α and that of the top of flagstaff is β. Prove that the height of the tower is 

`(h tan alpha)/(tan beta - tan alpha)`

Answer 1

Let height of tower AB = x

Height of flag staff CA = h.

Which makes the angles of elevation β and α at D.

In right ΔADB, we have

`tan alpha = (AB)/(DB) = x/(DB)`

∴ `DB = x/(tan alpha)`  ...(i)

And in right ΔCDB, we have

`tan beta = (CB)/(DB) = (h + x)/(DB)`

∴ `DB = (h + x)/(tan beta)`  ...(ii)

From (i) and (ii)

`(h + x)/tan beta = x/tan alpha`

`\implies` h tan α + x tan α = x tan β

`\implies` h tan α = x tan β – x tan α

`\implies` h tan α = x (tan β – tan α)

∴ `x = (h tan alpha)/(tan beta - tan alpha)`

Hence required height of tower = `(h tan alpha)/(tan beta - tan alpha)`