Find height of multistoried building

Question

The angle of depression of the top and bottom of an 8 m tall building from the top of a multi-storeyed building is 30 degrees and 60 degrees respectively find the height of multi-storeyed building and the distance b/w two buildings.

Answer 1

Given AB = 8 m be the height of the tall building
Hence CE = 8 m
Let the height of the multistoried building be CD = ‘h’ m
Therefore DE = CD – CE = (h – 8) m
Let BC = AE = x
In right ΔDCB, θ = 60°

$\Rightarrow \tan 60^{\cdot }=\dfrac {DC} {BC}$

$\Rightarrow \sqrt {3}=\dfrac {h} {x}$

$\therefore h=\sqrt {3}x\rightarrow \left( 1\right)$

In right ΔDEA, θ = 30°

$\begin{align*} & \Rightarrow \tan 30^{0}=\dfrac {DE} {EA}\\ & \Rightarrow \dfrac {1} {\sqrt {3}}=\dfrac {h-8} {x}\\ & \Rightarrow x\Rightarrow \sqrt {3}\left( h-8\right) \end{align*}$

$\begin{align*} & \Rightarrow \dfrac {h} {\sqrt {3}}=\sqrt {3}\left( h-8\right) \\ & \Rightarrow h=3\left( n-8\right) \\ & \Rightarrow h=3h-24\\ & \Rightarrow 2n=24\\ & \therefore h=12\end{align*}$

Thus the height of the multistoried building is 12 m