Question #40903

Let $P_1: x+y+z=5$ and $P_2: x+y-z=5$ be two planes. Let $\Pi$ be the plane passing through the line of intersection of $P_1$ and $P_2$, and perpendicular to the $xy$-plane. Let $\Pi'$ be the plane $x+y+5=0$. For any point $P(x,y,z)$ and a plane $A$, let $d_{alg}(P, A)$ denote the algebraic distance from $P$ to $A$ with respect to the unit normal vector of $A$ that has a positive $x$-component. Let $L$ be the $z$-axis, and $d(P, L)$ be the perpendicular distance from $P$ to $L$. A point $P(x,y,z)$ moves in the region bounded by $0 \le x \le 6$, $0 \le y \le 6$, and $0 \le z \le 4$ such that it satisfies the inequality: $\frac{d(P, L)^2 - 25}{d_{alg}(P, \Pi)} \le 2 d_{alg}(P, \Pi')$ What is the volume of the region representing the locus of $P$?