The longest straight rod that fits inside a cubical carton measures 15√3 cm. What is the carton's capacity, in cubic cm?

The longest rod lies along the space diagonal, whose length is a√3 for a cube of edge a. So a√3 = 15√3 gives $a=15$ cm and the capacity = $15^3=3375$ cubic cm. The trap $1350=6\times 15^2$ is the carton's surface area, an area offered where a volume was asked; $1125=15^3/3$ divides by 3 at the end, confusing the 3 under the root sign with a factor of the volume; $180=12\times 15$ is the total length of the twelve edges, a length rather than a volume.