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) 2013]

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 \[=l\,\,m\]

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

Now in \[\Delta ABC\]

\[\tan 60{}^\circ =\frac{AB}{BC}=\frac{x}{l}\]\[\Rightarrow \]\[\sqrt{3}=\frac{x}{l}\]

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

Now, in \[\Delta ABD\]

\[\tan 30{}^\circ =\frac{AB}{BD}=\frac{x}{l+80}\]

\[\Rightarrow \]   \[\frac{1}{\sqrt{3}}=\frac{x}{l+80}\]

\[\Rightarrow \]   \[l+80=\sqrt{3x}\]

Now, from Eq. (i),

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

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

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

\[\therefore \]Length of the field, \[l=40\,\,m\]