Question

A vertical lamp-post of height 9 meters stands at the corner of a rectangular field.The angle of elevation of its top from the farthest corner is ${30}^{\circ }$,while from another corner it is ${45}^{\circ }.$ The area of the field is

• A. $9\sqrt{2}meter{}^2$
• B. $81\sqrt{3}meter{}^2$
• C. $81\sqrt{2}meter{}^2$
• D. $9\sqrt{3}meter{}^2$

Answer 1

Answer: (C)

$81\sqrt{2}meter{}^2$

Consider rectangle $ABCD$ be the rectangular field. Let $PD$ be the height of vertical lamp post.

it is given that Height of the Pole $=PD=9$m

Now in $△PDA$

$tan{45}^o=\frac{PD}{DA}=\frac{9}{DA}\Rightarrow DA=9$

Now in $△PDB$

$tan{30}^o=\frac{9}{DB}\Rightarrow DB=9\sqrt{3}$

Also, we can see that $△DAB$ is Right Angled Triangle

$\therefore D{A}^2+A{B}^2=D{B}^2\Rightarrow 9^2+A{B}^2=(9\sqrt{3}{)}^2\Rightarrow A{B}^2=81\times 2\Rightarrow AB=9\sqrt{2}$

Now, Area of Rectangle $=DA\times AB=9\sqrt{2}\times 9=81\sqrt{2}$ $m^2$

Therefore, option (C) is the correct answer.