Lời giải:.png)
Trong \(\left( ABCD \right)\) gọi \(O=AC\cap BD.\)
Trong \(\left( SBD \right)\) gọi \(I=SO\cap MP.\)
Trong \(\left( SAC \right)\) gọi \(N=SC\cap AI.\)
Trong \(\left( SBD \right),\) qua M kẻ đường thẳng song song với BD cắt SO tại H, qua P kẻ đường thẳng song song với BD cắt SO tại K.
Gọi T là trung điểm NC.
Ta có: \(\frac{IH}{IK}=\frac{MH}{PK}=\frac{\frac{1}{2}BO}{\frac{2}{3}BO}=\frac{3}{4}.\)
\(HK=SO-SH-OK=SO-\frac{1}{2}SO-\frac{1}{3}SO=\frac{1}{6}SO.\)
\(\frac{IH}{3}=\frac{IK}{4}=\frac{IH+IK}{7}=\frac{\frac{1}{6}SO}{7}=\frac{1}{42}SO.\)
\(\frac{SI}{SO}=\frac{SH+IH}{SO}=\frac{\frac{1}{2}SO+\frac{1}{14}SO}{SO}=\frac{4}{7}.\)
\(\Rightarrow \frac{SN}{ST}=\frac{4}{7}.\)
\(\Rightarrow \frac{SN}{SC}=\frac{4}{10}=\frac{2}{5}.\)
\(\frac{{{V}_{S.AMNP}}}{{{V}_{S.ABCD}}}=\frac{1}{2}\left[ \frac{{{V}_{S.AMN}}}{{{S}_{S.ACB}}}+\frac{{{V}_{S.ANP}}}{{{V}_{S.ACD}}} \right]=\frac{1}{2}\left[ \frac{SM}{SB}.\frac{SN}{SC}+\frac{SP}{SD}.\frac{SN}{SC} \right]=\frac{1}{2}\left[ \frac{1}{2}.\frac{2}{5}+\frac{2}{5}.\frac{2}{3} \right]=\frac{7}{30}.\)
\({{V}_{ABCD.AMNP}}={{V}_{S.ABCD}}-{{V}_{S.AMNP}}=V-\frac{7}{20}V=\frac{23}{30}V.\)
