# A $1.5\ m$ tall boy is standing at some distance from a $30\ m$ tall building. The angle of elevation from his eyes to the top of the building increases from $30^o$ to $60^o$ as he walks towards the building. Find the distance he walked towards the building.

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Given:

A $1.5 \mathrm{~m}$ tall boy is standing at some distance from a $30 \mathrm{~m}$ tall building.

The angle of elevation from his eyes to the top of the building increases from $30^{\circ}$ to $60^{\circ}$ as he walks towards the building.

To do:

We have to find the distance he walked towards the building.

Solution:

Let $AB$ be the height of the boy and $CD$ be the height of the building.

From the figure,

$\mathrm{AB}=\mathrm{OP}=\mathrm{DE}=1.5 \mathrm{~m}, \angle \mathrm{CAE}=30^{\circ}, \angle \mathrm{COE}=60^{\circ}$

Let the distance he walked towards the building be $\mathrm{AO}=x \mathrm{~m}$.

This implies,

$\mathrm{CE}=30-1.5=28.5 \mathrm{~m}$

We know that,

$\tan \theta=\frac{\text { Opposite }}{\text { Adjacent }}$

$=\frac{\text { CE }}{OE}$

$\Rightarrow \tan 60^{\circ}=\frac{28.5}{OE}$

$\Rightarrow \sqrt3=\frac{28.5}{OE}$

$\Rightarrow OE=\frac{28.5}{\sqrt3} \mathrm{~m}$.........(i)

Similarly,

$\tan \theta=\frac{\text { Opposite }}{\text { Adjacent }}$

$=\frac{\text { CE }}{AE}$

$\Rightarrow \tan 30^{\circ}=\frac{28.5}{x+OE}$

$\Rightarrow \frac{1}{\sqrt3}=\frac{28.5}{x+OE}$

$\Rightarrow x+OE=28.5\sqrt3 \mathrm{~m}$

$\Rightarrow x=28.5\sqrt3-\frac{28.5}{\sqrt3} \mathrm{~m}$            [From (i)]

$\Rightarrow x=\frac{28.5\sqrt3(\sqrt3)-28.5}{\sqrt3} \mathrm{~m}$

$\Rightarrow x=\frac{28.5(3-1)}{\sqrt3}\times \frac{\sqrt3}{\sqrt3} \mathrm{~m}$

$\Rightarrow x=\frac{28.5(2)\times\sqrt3}{3} \mathrm{~m}$

$\Rightarrow x=9.5(2)\sqrt3=19\sqrt3 \mathrm{~m}$

Therefore, the distance he walked towards the building is $19\sqrt3 \mathrm{~m}$.

Updated on 10-Oct-2022 13:23:31