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, \[l=40\,\,m\] |
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