The angles of depression of the top and bottom of $ 8 \mathrm{~m} $ tall building from the top of a multistoried building are $ 30^{\circ} $ and $ 45^{\circ} $ respectively. Find the height of the multistoried building and the distance between the two buildings.

Given:

The angles of depression of the top and bottom of \( 8 \mathrm{~m} \) tall building from the top of a multistoried building are \( 30^{\circ} \) and \( 45^{\circ} \) respectively.

To do:

We have to find the height of the multistoried building and the distance between the two buildings.

Solution:  

Let $AB$ be the tall building and $CD$ be the height of the multistoried building.

Let point $D$ be the point of observation.

From the figure,

$\mathrm{AB}=8 \mathrm{~m}, \angle \mathrm{DAE}=30^{\circ}, \angle \mathrm{DBC}=45^{\circ}$

Let the height of the multistoried building be $\mathrm{CD}=h \mathrm{~m}$ and the distance between the two buildings be $\mathrm{AE}=\mathrm{BC}=x \mathrm{~m}$.

This implies,

$\mathrm{DE}=h-8 \mathrm{~m}$

We know that,

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

$=\frac{\text { DC }}{BC}$

$\Rightarrow \tan 45^{\circ}=\frac{h}{x}$

$\Rightarrow 1(x)=h$

$\Rightarrow h=x \mathrm{~m}$..........(i)

Similarly,

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

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

$\Rightarrow \tan 30^{\circ}=\frac{h-8}{x}$

$\Rightarrow \frac{1}{\sqrt3}=\frac{h-8}{h}$                  [From (i)]

$\Rightarrow h=(h-8)\sqrt3 \mathrm{~m}$

$\Rightarrow h=h\sqrt3-8\sqrt3 \mathrm{~m}$

$\Rightarrow h(\sqrt3-1)=8\sqrt3 \mathrm{~m}$

$\Rightarrow h=\frac{8\sqrt3}{\sqrt3-1} \mathrm{~m}$

$\Rightarrow h=\frac{8\sqrt3}{\sqrt3-1}\times\frac{\sqrt3+1}{\sqrt3+1} \mathrm{~m}$

$\Rightarrow h=\frac{8\sqrt3(\sqrt3+1)}{(\sqrt3)^2-1^2} \mathrm{~m}$

$\Rightarrow h=\frac{8(3+\sqrt3)}{3-1} \mathrm{~m}$

$\Rightarrow h=4(3+\sqrt3) \mathrm{~m}$

$\Rightarrow x=4(3+\sqrt3) \mathrm{~m}$

Therefore, the height of the multistoried building is $4(3+\sqrt3) \mathrm{~m}$ and the distance between the two buildings is $4(3+\sqrt3) \mathrm{~m}$.   

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Updated on: 10-Oct-2022

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