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 $30$ degrees, while from another corner it is $45$ degrees. The area of the field is

• A. $81$ sq. meters
• B. $81\sqrt{2}$ sq. meters
• C. $9\sqrt{2}$ sq. meters
• D. none of these

Answer 1

The correct option is B

$81\sqrt{2}$ sq. meters

Explanation for the given option:

Step-1: Draw the figure and find the side of the rectangle:

Given, Height of lamp-post of height $9$ m.

In $∆ABC$,

$\Rightarrow$$BC=\frac{9}{\tan \left(30°\right)}$

$\Rightarrow$$BC=9\sqrt{3}m$

In $∆ABD$,

$\Rightarrow$$BD=\frac{9}{\cot \left(30°\right)}$

$\Rightarrow$$BD=9m$

Step-2: Apply Pythagoras theorem In $∆BCD$, $DC=\sqrt{{\left(BD\right)}^2+{\left(BC\right)}^2}$.

$\Rightarrow$$BC=\sqrt{{\left(9\sqrt{3}\right)}^2-{\left(9\right)}^2}$

$\Rightarrow$$BC=9\sqrt{2}m$

Step-3: Finding area of field(rectangle) by using formula $\left(length\times width\right)$.

$\Rightarrow$ $A=9\sqrt{2}\times 9$

$\Rightarrow$ $A=81\sqrt{2}$sq. meters

Hence, option $\left(B\right)$ is correct.