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 |
\[l+80=\sqrt{3}\,(\sqrt{3}l)\] |
\[\Rightarrow \] \[l+80=3l\]\[\Rightarrow \]\[80=3l-l\] |
\[\Rightarrow \] \[l=\frac{80}{2}=40\,\,m\] |
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