2016 AMC 12B Problem 23

Below is the professionally curated solution for Problem 23 of the 2016 AMC 12B, from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2016 AMC 12B solutions, or check the answer key.

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Concepts:3D geometry volume power scaling of length, area, and volume

Difficulty rating: 2270

23.

What is the volume of the region in three-dimensional space defined by the inequalities $|x|+|y|+|z|\le1$ and $|x|+|y|+|z-1|\le1?$

$\dfrac16$

$\dfrac13$

$\dfrac12$

$\dfrac23$

$1$

Solution:

The region $|x|+|y|+|z|\le1$ is a regular octahedron with vertices at $(\pm1,0,0),(0,\pm1,0),(0,0,\pm1),$ whose volume is $2\cdot\tfrac13\cdot(\sqrt2)^2\cdot1=\tfrac43.$ The second region is the same octahedron shifted up by $1.$ Their intersection is bounded by another regular octahedron with diagonals of length $1,$ half the linear dimensions of the first, so its volume is $\left(\tfrac12\right)^3\cdot\tfrac43=\tfrac16.$

Thus, the correct answer is A.

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