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Cuboid Perimeters To Volume

Problem

For any given cuboid it is possible to measure up to three different perimeters. For example, one perimeter could be measured this way.

Given that cuboid A has perimeters 12, 16, and 20, and cuboid B has perimeters 12, 16, and 24, which cuboid has the greatest volume?


Solution

Let the dimensions of a cuboid be $x, y, z$. The perimeters will be $2(x+y), 2(x+z),$ and $2(y+z)$, so dividing each of the perimeters by 2 will give $x+y, x+z,$ and $y+z$ respectively.

The sum of these three terms will give $2x+2y+2z$, leading to $x+y+z$. Thus by subtracting each of $x+y, x+z,$ and $y+z$ we will be able to obtain $z, y,$ and $x$.

Cuboid$2(x+y)$$2(x+z)$$2(y+z)$$x+y$$x+z$$y+z$
A1216206810
B1216246812

Cuboid$2(x+y+z)$$x+y+z$$x$$y$$z$$xyz$
A241224648
B261315735

Surprising as it may seem, cuboid A has the greater volume.

Problem ID: 357 (26 Aug 2009)     Difficulty: 3 Star

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