The length of the longest rod that can be put in a room of dimensions $10\text{ m}\times 10\text{ m}\times 5\text{ m}$ is

The length of the longest rod that can be fit into a rectangular room corresponds to its maximum inner space diagonal length ($D$). The formula for a cuboid's diagonal is:
$$D=\sqrt{l^2+w^2+h^2}$$

Given dimensions $l=10\text{ m}$, $w=10\text{ m}$, and $h=5\text{ m}$:

$$D=\sqrt{10^2+10^2+5^2}=\sqrt{100+100+25}=\sqrt{225}=15\text{ m}$$