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Question 8
A vertical tower stands on a horizontal plane and is surmounted by a vertical flagstaff of height "h" units. At a point on the plane, the angles of elevation of the bottom and the top of the flagstaff are α and β respectively. Prove that the height of the tower is (h tan αtan βtan α).

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Solution

Let the height of the tower be H and OR = x

Given that, height of flag staff = h = FP and PRO=α,FRO=β

Now, in ΔPRO, tan α=PORO=Hx

x=Htan α ....(i)

And in ΔFRO, tan β=FORO=FP+POROtan β=h+Hx

x=h+Htan β(ii)

From Eqs.(i) and (ii),

Htan α=h+Htan β

H tan β=h tan α+H tan α

H tan βH tan α=h tan α

H(tan βtan α)=h tanαH=h tan αtan βtan α

Hence, the required height of tower is h tan αtan βtan α.



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