Heights and Distances Practice Test 2

Question : 13 of 24

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At a point on a level plane a tower subtends an angle θ and a flag-staff a ft. in length at the top of the tower subtends an angle ϕ. The height of the tower is:

${a\sinθ\cosϕ}/{\cos (θ+ϕ)}$
${a\sinθ\cos(θ+ϕ)}/{\sinϕ}$
${a\cos(θ+ϕ)}/{\sinθ\sinϕ}$
None of these

Validate

Solution:

Let OP be the tower of height h (say) and PQ the flag-staff of height a, such that
∠OAP = θ and ∠ PAQ = ϕ
In ΔOAP and ΔOAQ
OA = OP cot θ = h cot θ
and OA = OQ cot (θ + ϕ) = (h + a) cot (θ + ϕ)
∴ h cot θ = (h + a) cot (θ + ϕ)

⇒h={a\cot(θ+ϕ)}/{\cotθ-\cot(θ+ϕ)}

={a\sinθ\cos(θ+ϕ)}/{\sinϕ}

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