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 length of the field is                                                         [SSC (CGL) 2012]

A) \[20\,\,m\]                     

B) \[40\sqrt{2}\,\,m\]

C) \[40\,\,m\]

D) \[20\sqrt{2}\,\,m\]

Correct Answer: C

Solution :

Let the length of football field be \[l\,\,m.\]

From the figure,

Height of the pole \[=x\,\,m\]

\[\therefore \]In\[\Delta ABC,\]\[\tan 60{}^\circ =\frac{x}{l}\]\[\Rightarrow \]\[\sqrt{3}=\frac{x}{l}\]

\[\Rightarrow \]   \[x=\sqrt{3}l\]                            ... (i)

Now, in \[\Delta ABD\]

\[\tan 30{}^\circ =\frac{x}{l+80}\]\[\Rightarrow \]\[\frac{1}{\sqrt{3}}=\frac{x}{l+80}\]

\[\Rightarrow \]   \[l+80=\sqrt{3}x\]

Now, from Eq. (i), we get

\[l+80=\sqrt{3}\,(\sqrt{3}l)\]

\[\Rightarrow \]   \[l+80=3l\]\[\Rightarrow \]\[80=3l-l\]

\[\Rightarrow \]   \[l=\frac{80}{2}=40\,\,m\]