Question

The angle of elevation of the top of a vertical pole when observed from each vertex of a regular hexagon is $\frac{\pi }{3}$ if the area of the circle circum scribing the hexagon be $Am$, then the height of the tower is

• A. $\frac{2A}{\sqrt{3\pi }}m$
• B. $\frac{A}{\sqrt{3\pi }}m$
• C. $2\sqrt{\frac{A}{3\pi }}m$
• D. $\sqrt{\frac{3A}{\pi }}m$

Answer 1

The correct option is D $\sqrt{\frac{3A}{\pi }}m$

Given that:

Angle of elevation from vertex of a regular hexagon$=\frac{\pi }{3}$

Area of the circle circumscribing the hexagon $=Am$

To find:

Height of the tower $=?$

Solution:

Let $r$ be the radius of circle and $h$ be the height of tower.

Area of circle $=\pi r^2$

$\Rightarrow A=\pi r^2$

$\Rightarrow r=\sqrt{\frac{A}{\pi }}$

Now, $\tan \left(\frac{\pi }{3}\right)=\frac{h}{r}$

or, $h=r\tan \left(\frac{\pi }{3}\right)$

or, $h=\sqrt{\frac{A}{\pi }}\times \sqrt{3}$

or, $h=\sqrt{\frac{3A}{\pi }}$