Question

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 top of the flag staff are alpha and beta respectively. prove that the height of the tower is h tan alpha/tan beta - tan alpha

Answer 1

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Answer 2

Answer:

The height of the tower is h tan alpha/tan beta - tan alpha is proved

Solution:

Proof:

The diagram for this sum is attached below

Let AB be the tower and BC be the flagstaff. Let OA = x metre, AB = ‘y’ metre and  BC = ‘h’ metre.

In right ∆OAB,

$\tan \alpha=\frac{A B}{O A}$

$\tan \alpha=\frac{y}{x}$

On cross-multiplication we get,

$\mathrm{x}=\frac{y}{\tan \alpha}$   → (1)

In right ∆OAC

$\begin{array}{l}{\tan \beta=\frac{A C}{O A}} \\\\ {\tan \beta=\frac{y+h}{x}}\end{array}$  

$\mathrm{x}=\frac{y+h}{\tan \beta}$  → (2)

Equating equation 1 and 2 we get,

$\frac{y}{\tan \alpha}=\frac{y+h}{\tan \beta}$

$\begin{array}{l}{\underset{~} \tan \beta=y \tan \alpha+h \tan \alpha} \\ {\underset{~}{y \tan \beta-y \tan \alpha}=h \tan \alpha} \\ {y(\tan \beta-\tan \alpha)=h \tan \alpha} \\ {y=\frac{\text { htan\alpha }}{(\tan \beta-\tan \alpha)}=A B}\end{array}$

Thus the height of the tower is h tan alpha/tan beta - tan alpha is proved