question_answer

A man standing in one corner of a square football field observes that the angle subtended by a pole in the corner just diagonally opposite to this corner is \[60{}^\circ .\] When he retires 80 m from the corner, along the same straight line, he finds the angle to be\[30{}^\circ .\]The side of the field is

A) \[20\,\,m\]

B)         \[40\sqrt{2}\,\,m\]

C) \[40\,\,m\]         

D)  \[20\sqrt{2}\,\,m\]

Correct Answer: D

Solution :

From the figure, Let the diagonal of football field be = l m and height of the pole = x m \[\therefore \]In \[\Delta \,\,ABC,\tan 60{}^\circ =\frac{x}{l}\sqrt{3}=\frac{x}{l}\] \[\Rightarrow \]               \[x=\sqrt{3}l\]                ?(i) Now. in \[\Delta ABD,\,\,\,\tan 30{}^\circ =\frac{x}{l+80}\] \[l+80=\sqrt{3}x\] Now, from Eq. (i), \[l+80=\sqrt{3}(\sqrt{3}l)\]             \[\Rightarrow \]               \[l+80=3l\] \[\therefore \]\[l=\frac{80}{2}=40\,\,cm=\]diagonal of square field Hence, side of the square field is given by \[=\frac{\text{Diagonal}}{\sqrt{2}}=\frac{40}{\sqrt{2}}=20\sqrt{2}\,\,m\]