Question

The angles of depression of the top and the bottom of a $10m$ tall building observed from the top of a tower are ${30}^0$ and ${45}^0$ respectively. Find the height of the tower (in $m$)

• A. $20+5\sqrt{2}$
• B. $12+10\sqrt{2}$
• C. $15+5\sqrt{3}$
• D. $18+3\sqrt{3}$

Answer 1

The correct option is C $15+5\sqrt{3}$

Let the height of the tower be $H$

we can see that, $BCDE$ is a rectangle

so, $BE=CD=10\&BC=DE$

Also, we can see that, $H=AC+CD\Rightarrow H=AC+10\Rightarrow AC=H-10$

Now, in $△ADE$

$\tan {45}^o=\frac{AD}{DE}\Rightarrow 1=\frac{AD}{DE}\Rightarrow AD=DE\Rightarrow DE=H$

$\therefore BC=DE=H$

Now, in $△ABC$

$\tan {30}^o=\frac{AC}{BC}\Rightarrow \frac{1}{\sqrt{3}}=\frac{H-10}{H}\Rightarrow H=\sqrt{3}H-10\sqrt{3}\Rightarrow 10\sqrt{3}=(\sqrt{3}-1)H\Rightarrow H=\frac{10\sqrt{3}}{\sqrt{3}-1}=\frac{10\sqrt{3}\ast (\sqrt{3}+1)}{(\sqrt{3}-1)(\sqrt{3}+1)}=5\sqrt{3}(\sqrt{3}+1)=15+5\sqrt{3}$

Therefore, Answer is $15+5\sqrt{3}$