question_answer

A pyramid is formed by cutting off a corner of a cube whose edge is two metres by a plane which bisects its three co-terminus edges. The volume of the pyramid is

A)  \[1\,{{m}^{3}}\]              

B)         \[\frac{1}{2}\,{{m}^{3}}\]

C)  \[\frac{1}{3}\,{{m}^{3}}\]                           

D)         \[\frac{1}{6}\,{{m}^{3}}\]  

Correct Answer: D

Solution :

\[OA=O{{A}^{2}}-A{{P}^{2}}=\frac{1}{3}\] \[AB=BC=CA=\sqrt{2}\] \[AQ=\sqrt{2}\,\,\sin \,{{60}^{o}}\]                 \[=\sqrt{\frac{3}{2}}\] \[AP=\frac{2}{3}.\frac{\sqrt{3}}{2}=\frac{\sqrt{2}}{3}\] and        \[OP=O{{A}^{2}}-A{{P}^{2}}=\frac{1}{3}\] \[\therefore \]  Required volume                 \[=\frac{1}{3}\times \]Area of \[\Delta \,ABC\times OP\]                 \[=\frac{1}{6}\,{{m}^{3}}\]