Question

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 `((h tan α)/(tan β - tan α))`.

Answer 1

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

tan α = `"PO"/"RO" = "H"/x`   ...`[∵ tan θ = "Perpendicular"/"base"]`

⇒ x = `"H"/tan α`  ...[Equation 1]

And in ∆FRO, we have

tan β = `"FO"/"RO" = ("FP" + "PO")/"RO"`

tan β = `("h" + "H")/x`

⇒ x = `("h" + "H")/tan β`  ...[Equation 2]

Comparing equation 1 and equation 2,

⇒ `"H"/tan α = ("h" + "H")/tan β`

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

⇒ H = `("h" tan α)/(tan β - tan α)`

Hence, proved.