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A flag-staff of height h stand on the top of a school building

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A flag-staff of height h stand on the top of a school building. If the angles of elevation of the and bottom of the flag-staff have measure $α$ and $β$ are respectively from a point on the ground, prove that the height of the building is $\frac{h tan β}{tan α - tan β}$

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A flag-staff of height h stand on the top of a school building. If the angles of elevation of the and bottom of the flag-staff have measure

α

and

β

are respectively from a point on the ground, prove that the height of the building is

\frac{h tan β}{tan α - tan β}

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Text solutionVerified

Let AB be the tower and (A, B)

the flag staff, CA

=

Let D be observation point, such that angle of elevation of top & c bottom, be

α

β

respectively.

Δ A B D

tan β =

\frac{A B}{B D}

B D = A B cot β

…………..

(1)

Δ CB D

tan α

= \frac{CB}{B D}

B D = CB cot α [∴ CB = C A + A B = h + A B]

B D = (h + a B) cot α

………

(2)

Equating

(1)

(2)

equation,

A B cot β = (h + A B) cot α

A B cot β = h cot α + A B cot α

A B (cot β - cot α) = h cot α

A B = \frac{h cot α}{cot β - cos α}

A B = \frac{\frac{h}{tan α}}{\frac{tan α - tan β}{tan β \cdot tan α}}

A B = h (\frac{tan β}{tan α - tan β})

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Question Text	A flag-staff of height h stand on the top of a school building. If the angles of elevation of the and bottom of the flag-staff have measure $α$ and $β$ are respectively from a point on the ground, prove that the height of the building is $\frac{h tan β}{tan α - tan β}$
Answer Type	Text solution:1
Upvotes	151