A vertical tower stands on a horizontal plane and is surmounted by a vertical flag staff of height h. At a point on the plane, the angles of elevation of the bottom and the top of the flag staff are α and β respectively. Prove that the height of the tower is .

Given that a vertical flag staff of height h is surmounted on a vertical tower of height H(say),

such that FP = h and FO = H.

The angle of elevation of the bottom and top of the flag staff on the plane is ∠PRO = α and ∠FRO = β respectively.

In ∆PRO, we have

And in ∆FRO, we have

Comparing eq. 1 and eq. 2,

Solving for H,

⇒ H tan β = (h+H) tan α

⇒ H tan β – H tan α = h tan α

⇒ H (tan β – tan α) = h tan α

Hence, proved.

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