The breadth of a room is twice its height,

Suppose that the breadth of the room $=\mathrm{x}$ dm

Since breadth is twice the height, breadth $=2 \times$ height

So, height of the room $=\frac{\text { breadth }}{2}=\frac{x}{2}$

Also, it is given that the breadth is half the length.

So, breadth $=\frac{1}{2} \times$ length

i. e., length $=2 \times$ breadth $=2 \times \mathrm{x}$

Since volume of the room $=512 \mathrm{cu} \mathrm{dm}$, we have :

Volume of a cuboid $=$ length $\times$ breadth $\times$ height

$\Rightarrow 512=2 \times \mathrm{x} \times \mathrm{x} \times \frac{\mathrm{x}}{2}$

$\Rightarrow 512=\mathrm{x}^{3}$

$\Rightarrow \mathrm{x}=\sqrt[3]{512}=8 \mathrm{dm}$

Hence, length of the room $=2 \times \mathrm{x}=2 \times 8=16 \mathrm{dm}$

Breadth of the room $=x=8 \mathrm{dm}$

Height of the the room $=\frac{\mathrm{x}}{2}=\frac{8}{2}=4 \mathrm{dm}$