Question

The angle of elevation of an electric pole from a point $A$ is ${60}^{\circ }$ and from a point $B$ towards the pole on the line joining the foot of the pole to the point is ${75}^{\circ }$. If the distance $AB=a$, then the height of the pole is:

• A. $\frac{a(3+2\sqrt{3})}{2}$
• B. $a(4+2\sqrt{3})$
• C. $\frac{a(2+\sqrt{3})}{2}$
• D. $\frac{a(2\sqrt{3}-3)}{2}$

Answer 1

The correct option is A $\frac{a(3+2\sqrt{3})}{2}$

Let OP be the electric pole

$\tan {75}^{\circ }=(2+\sqrt{3})$

From $△OPA,$

$\tan {75}^{\circ }=\frac{h}{b}$

$\Rightarrow b=\frac{h}{2+\sqrt{3}}$ ...(i)

From $△OPB,$
$\tan {60}^{\circ }=\frac{h}{a+b}$

$⟹\sqrt{3}=\frac{h}{a+b}$

$⟹h=\sqrt{3}(a+b)$ ...(ii)

From (i) and (ii), we get

$h=\sqrt{3}(a+\frac{h}{(2+\sqrt{3})})$

$\Rightarrow h\left(\frac{2+\sqrt{3}-\sqrt{3}}{(2+\sqrt{3})}\right)=a\sqrt{3}$

$\Rightarrow h=\frac{a(3+2\sqrt{3})}{2}$