A packaging company is going to make closed boxes, with square bases, that hold 2,197 cubic centimeters. What are the dimensions of the box that can be built with the least material
A) $6.5 \times 6.5 \times 52$ cm
B) $3.25 \times 3.25 \times 208$ cm
C) $13 \times 13 \times 13$ cm
D) $1 \times 1 \times 2,197$ cm
E) $26 \times 26 \times 3.25$ cm

Question

A packaging company is going to make closed boxes, with square bases, that hold 2,197 cubic centimeters. What are the dimensions of the box that can be built with the least material   
A) $6.5 \times 6.5 \times 52$ cm
B) $3.25 \times 3.25 \times 208$ cm
C) $13 \times 13 \times 13$ cm
D) $1 \times 1 \times 2,197$ cm
E) $26 \times 26 \times 3.25$ cm

Quizplus · Accepted Answer

The volume of the box is given as 2,197 cubic centimeters, which is equal to $13^3$. For a box with a square base to use the least material (i.e., have the smallest surface area), its dimensions should be as close to a cube as possible. Among the given options, $13 \times 13 \times 13$ cm is the only cubic dimension, making it the most material-efficient configuration.

A Packaging Company Is Going to Make Closed Boxes, with Square