Question

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 $\alpha$ and $\beta$ are respectively from a point on the ground, prove that the height of the building is $\frac{h\tan \beta }{\tan \alpha -\tan \beta }$

Answer 1

Let AB be the tower and (A, B)

the flag staff, CA$=$hm

Let D be observation point, such that angle of elevation of top & c bottom, be $\alpha$ & $\beta$ respectively.

In $\Delta ABD$,

$\tan \beta =$$\frac{AB}{BD}$

$BD=AB\cot \beta$ …………..$\left(1\right)$

In $\Delta CBD$,

$\tan \alpha$ $=\frac{CB}{BD}$

$BD=CB\cot \alpha [\therefore CB=CA+AB=h+AB]$

$BD=(h+aB)\cot \alpha$ ………$\left(2\right)$

Equating $\left(1\right)$ & $\left(2\right)$ equation,

$AB\cot \beta =(h+AB)\cot \alpha$

$AB\cot \beta =h\cot \alpha +AB\cot \alpha$

$AB(\cot \beta -\cot \alpha )=h\cot \alpha$

$AB=\frac{h\cot \alpha }{\cot \beta -\cos \alpha }$

$AB=\frac{\frac{h}{\tan \alpha }}{\frac{\tan \alpha -\tan \beta }{\tan \beta \cdot \tan \alpha }}$

$AB=h\left(\frac{\tan \beta }{\tan \alpha -\tan \beta }\right)$,