Question

A vertical tower stands on a horizontal plan 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 tanα/ tanβ - tanα).

Answer with full explanation...!!!
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Answer 1

let the height of the tower = x m

∴ tanα = DB/BC = x / b

b = x/tanα                       ----(i)

∴ tanβ = AB/BC = (x + h)/b

b = (x + h)/tanβ                 ---------(ii)

eq (i) = eq. (ii)

x/tanα = (x + h)/tanβ

x = (x + h)tanα/tanβ

x.tanβ = x.tanα + h.tanα

x.tanβ - x.tanα =  h.tanα

x(tanβ - tanα) =  h.tanα

x =  h.tanα /(tanβ - tanα)

Answer 2

Question :

A vertical tower stands on a horizontal plan 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 tanα/ tan β - tan α).

Given :

Height of flag staff = h = FP and angle PRQ = α, FRO = β.

Solution :

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

Now, In ΔPRO, tan α = PO/RO = H/x

≈ x = H/tan α

______________(1)

and in ΔFRO, tan β = FO/RO = FP+PO/RO

tan β = h+H/x

≈ x = h+H/tan β

______________(2)

From eqs. (1) & (2),

H/tan α = h+H/tan β

≈ H tan β = h tan α +H tan α

≈ H tan β - H tan α = h tan a

≈ H (tan β - tan α) = h tan α

≈ H = h tan α/ tan β - tan α

Hence the required height of tower is

h tan α/ tan β - tan α

Hence Proved...!!!!