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प्रश्न
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))`.
उत्तर
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.
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