## 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 | $2(x+y)$ | $2(x+z)$ | $2(y+z)$ | $x+y$ | $x+z$ | $y+z$ |
---|---|---|---|---|---|---|

A | 12 | 16 | 20 | 6 | 8 | 10 |

B | 12 | 16 | 24 | 6 | 8 | 12 |

Cuboid | $2(x+y+z)$ | $x+y+z$ | $x$ | $y$ | $z$ | $xyz$ |
---|---|---|---|---|---|---|

A | 24 | 12 | 2 | 4 | 6 | 48 |

B | 26 | 13 | 1 | 5 | 7 | 35 |

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

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