The breadth of a room is twice its height, one half of its length and the volume of the room is 512 cu. dm. Find its dimensions.

#### Solution

\[\text { Suppose that the breadth of the room = x dm }\]

\[\text { Since breadth is twice the height, breadth }= 2 \times \text { height }\]

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

\[\text { Also, it is given that the breadth is half the length .} \]

\[\text { So, breadth }= \frac{1}{2} \times \text { length }\]

\[\text { i . e . , length }= 2 \times \text { breadth } = 2 \times x\]

\[\text { Since volume of the room = 512 cu dm, we have } \]

\[\text { Volume of a cuboid = length } \times\text { breadth } \times \text { height }\]

\[ \Rightarrow 512 = 2 \times x \times x \times \frac{x}{2}\]

\[ \Rightarrow 512 = x^3 \]

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

\[\text { Hence, length of the room }= 2 \times x = 2 \times 8 = 16 dm \]

\[\text { Breadth of the room = x = 8 dm }\]

\[\text { Height of the the room } = \frac{x}{2}=\frac{8}{2} = 4 dm\]