Raj stands in a corner of his square farm. Angle of elevation of a scarecrow placed in diagonally opposite corner is 60°. He starts walking backwards in a straight line and after 80ft he realizes that angle of elevation of the scarecrow now is 30°. What is area of the field?

Answer: Option D

Solution

Answer: Option D

Solution:

$$\begin{array}{rl}& \tan 60^{\circ }=\sqrt{3}=\frac{PQ}{QR}\\ & \therefore PQ=\sqrt{3}QR\\ & \tan 30^{\circ }=\frac{1}{\sqrt{3}}\\ & =\frac{PQ}{SQ}\\ & =\frac{PQ}{80+QR}\\ & \therefore 80+QR=\sqrt{3}PQ\\ & \therefore 80+QR=3QR\\ & \therefore QR=40\text{ft}\text{.}\end{array}$$
If we read carefully, we see that Raj (Point R) and the scarecrow (Point Q) are in diagonally opposite corners.
So QR is a diagonal of the square farm.
Diagonal of square = side x$$\sqrt{2}$$
$$\begin{array}{rl}& \therefore 40=\text{side}\text{x}\sqrt{2}\\ & \therefore \text{side}=\frac{40}{\sqrt{2}}\\ & \therefore \text{Area}={\left(\text{side}\right)}^2\\ & ={\left(\frac{40}{\sqrt{2}}\right)}^2\\ & =\left(\frac{1600}{2}\right)\\ & =800\text{sq}\text{.}\text{ft}\text{.}\end{array}$$