A vertical tower stands on a horizontal plane and is surmounted by a vertical flagstaff of height $h$. At a point on the plane, the angles of elevation of the bottom and the top of the flag staff are $\alpha$ and $\beta$, respectively. Then find the height of the tower.
Given: A vertical tower stands on a horizontal plane and is surmounted by a vertical flagstaff of height $h$. At a point on the plane, the angles of elevation of the bottom and the top of the flag staff are $\alpha$ and $\beta$, respectively.
To do: To find the height of the tower.
Solution:
Let height be $y \vartriangle OAC$
$tan\beta =\frac{CA}{OA}$
$tan\beta =\frac{y+h}{x}$ $( y+h)=CA=AB+BC,\ Let OA=x$
$x=( \frac{y+h}{tan\beta})$
Consider $\vartriangle OAB$
$tan\alpha =\frac{y}{x}$
$x=\frac{y}{tan\alpha }$
$\frac{y}{tan\alpha }=\frac{y+h}{tan\beta}$
$y.tan\beta =y.tan\alpha +h.tan\alpha $
$y.tan\beta −y.tan\alpha=h.tan\alpha $
$y(tan\beta −tan\alpha )=h.tan\alpha $
$y=\frac{h.tan\alpha}{tan\beta −tan\alpha }$
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