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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)`
Solution
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)`
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