csat-suneung· South-Korea· csat__math-science4 marks
In coordinate space, consider the triangle ABC with vertices $\mathrm { A } ( 54,0,0 ) , \mathrm { B } ( 0,27,0 ) , \mathrm { C } ( 0,0,27 )$ on the plane $x + 2 y + 2 z = 54$. A point $\mathrm { P } ( x , y , z )$ is in the interior of triangle ABC. Let Q be the orthogonal projection of P onto the $xy$-plane, R be the orthogonal projection of P onto the $yz$-plane, and S be the orthogonal projection of P onto the $zx$-plane. When $\overline { \mathrm { QR } } = \overline { \mathrm { QS } }$, find the maximum volume of the tetrahedron QPRS. [4 points]
In coordinate space, consider the triangle ABC with vertices $\mathrm { A } ( 54,0,0 ) , \mathrm { B } ( 0,27,0 ) , \mathrm { C } ( 0,0,27 )$ on the plane $x + 2 y + 2 z = 54$. A point $\mathrm { P } ( x , y , z )$ is in the interior of triangle ABC. Let Q be the orthogonal projection of P onto the $xy$-plane, R be the orthogonal projection of P onto the $yz$-plane, and S be the orthogonal projection of P onto the $zx$-plane. When $\overline { \mathrm { QR } } = \overline { \mathrm { QS } }$, find the maximum volume of the tetrahedron QPRS. [4 points]