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# An observer $ 1.5 $ metres tall is $ 20.5 $ metres away from a tower 22 metres high. Determine the angle of elevation of the top of the tower from the eye of the observer.

Given:

An observer \( 1.5 \) metres tall is \( 20.5 \) metres away from a tower 22 metres high.

To do:

We have to determine the angle of elevation of the top of the tower from his eye.

Solution:

Let $AB$ be the tower and $CD$ be the observer who is $20.5\ m$ away from the tower.

From the figure,

$AB = 22\ m, CD=1.5\ m, AC=20.5\ m$

This implies,

$AE=CD=1.5\ m, DE=AC=20.5\ m$ and $BE=22-1.5=20.5\ m$

Let \( \theta \) be the angle of elevation of the top of the tower from the eye of the observer.

In $\Delta \mathrm{BDE}$,

$\tan \theta=\frac{\text { Perpendicular }}{\text { Base }}$

$\Rightarrow \tan \theta=\frac{\mathrm{BE}}{\mathrm{DE}}$

$=\frac{20.5}{20.5}$

$=1$

$=\tan 45^{\circ}$ [Since $\tan 45^{\circ}=1$]

$\Rightarrow \theta=45^{\circ}$

Therefore, the angle of elevation of the top of the tower from his eye is $45^{\circ}$.

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