# mathschallenge.net

## 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

Only Show Problem
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 15, "mathjax_display_tex": 4, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.8987846374511719, "perplexity_flag": "middle"}