Question

A vertical pole of height $10$ meters stands at one corner of a rectangular field. The angle of elevation of its top from the farthest corner is ${30}^{\circ }$, while that from another corner is ${60}^{\circ }$. The area (in $m^2)$ of rectangular field is

• A. $\frac{200\sqrt{2}}{3}$
• B. $\frac{400}{\sqrt{3}}$
• C. $\frac{200\sqrt{2}}{\sqrt{3}}$
• D. $\frac{400\sqrt{2}}{\sqrt{3}}$

Answer 1

The correct option is A $\frac{200\sqrt{2}}{3}$

$\tan {30}^{\circ }=\frac{10}{AC}=\frac{1}{\sqrt{3}}$

$\therefore \sqrt{l^2+b^2}=AC=10\sqrt{3}\text{cm}$ -----------(i)

$\tan {60}^{\circ }=\frac{10}{DC}=\sqrt{3}$

$\therefore DC=\frac{10}{\sqrt{3}}=l$

In traingle DCE

$l^2=\frac{100}{3}$ -----------(ii)

$b^2=300-\frac{100}{3}$

$b^2=\frac{800}{3}$

$l^2.b^2=\frac{100}{3}\times \frac{800}{3}=\frac{80000}{9}$

$Area=lb=\frac{\sqrt{80000}}{\sqrt{9}}=\frac{200\sqrt{2}}{3}{\text{m}}^2$.