Question

From the top of a house $32$ metres high, if the angle of elevation of the top of a tower is ${45}^o$ and the angle of depression of the foot of the tower is ${30}^o$, then the height of the tower is

• A. $\frac{32}{\sqrt{3}}(\sqrt{3}+1)$ metres
• B. $32(\sqrt{3}+1)$ metres
• C. $32\sqrt{3}$ metres
• D. $\frac{32}{3}(\sqrt{3}+1)$ metres

Answer 1

The correct option is B $32(\sqrt{3}+1)$ metres

Let $AD$ be the height of the house

$AD=h=32m$

$BC$ be the height of the tower

Let, $BC=x+y$

As, $ABED$ is a rectangle,

$x=h=32m$

In $△BDE,$

$\tan {30}^0=\frac{BE}{DE}$

$\Rightarrow \frac{1}{\sqrt{3}}=\frac{32}{DE}$

$\Rightarrow DE=32\sqrt{3}$

and In $△DEC,DE=32\sqrt{3}$

In $△DEC$,

$\tan {45}^0=\frac{CE}{DE}$

$\Rightarrow 1=\frac{y}{32\sqrt{3}}$

$\Rightarrow y=32\sqrt{3}$

Now,$BC=x+y=(32+32\sqrt{3})m=32(1+\sqrt{3})m$