A vertical tower stands on a horizontal plane 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 $\frac{h t a n α}{t a n β - t a n α}$

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Solution

Let AB be the tower and BC be the flag-staff. Let O be a point on the plane containing the foot of the tower such that the angles of elevation of the bottom B and top C of the flag-staff at O are $α$ and $β$ respectively. Let OA = x metres, AB = y metres and and BC = h metres.

In $Δ O A B$ , we have

$t a n α = \frac{A B}{O A}$

$\Rightarrow t a n α = \frac{y}{x}$

$\Rightarrow x = \frac{y}{t a n α}$ ........ (i)

$\Rightarrow x = y c o t α$

In $Δ O A C$ , we have

$t a n β = \frac{y + h}{x}$

$\Rightarrow x = \frac{y + h}{t a n β}$

$\Rightarrow x = (y + h) c o t β$ ....... (ii)

On equating the values of x given in equations (i) and (ii), we get

$\Rightarrow y c o t α = (y + h) c o t β$

$\Rightarrow (y c o t α - y c o t β) = h c o t β$

$\Rightarrow (y c o t α - y c o t β) = h c o t β$

$\Rightarrow y = \frac{h c o t β}{c o t α - c o t β}$

$\Rightarrow y = \frac{\frac{h}{t a n β}}{\frac{1}{t a n α} - \frac{1}{t a n β}} = \frac{h t a n α}{t a n β - t a n α}$

Hence, the height of the tower is $\frac{h t a n α}{t a n β - t a n α}$

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