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Question #41702

Difficulty: 8/10 | 🟢 Free

Let $R$ be the region in 3D space consisting of all points $P(x,y,z)$ that satisfy the following conditions:

If the volume of the region $R$ is $V$ , find the value of $3V$ .

Practice problems that bridge the gap between foundational theory and advanced practical application.

Word Problem Modeling|Distance from point to plane|Octant Sign Convention

This question tests the 'Octant Sign Convention' and the dual-bound nature of the distance formula.

A common lethal mistake is defaulting to the first octant ( $x,y,z \ge 0$ ), which yields a completely different constraint and a volume of 1/3.

Another high-frequency trap is dropping the absolute value in the distance inequality (solving $d \le 1$ instead of $|d| \le 1$ ).

This causes the student to miss the lower bound constraint, subtracting only one tetrahedron and arriving at the trap answer of 23.

Proper algebraic substitution for the specific octant is required to correctly partition the feasible region.