Question

A solid cube of side $12cm$ is cut into eight cubes of equal volume. What will be the side of the new cube? Also, find the ratio between their surface areas.

Answer 1

We know volume of cube $={\left(side\right)}^3$

$={\left(12\right)}^3$

$=1728m^3$

Surface area of cube$=6a^2$

$=6\times {\left(12\right)}^2-------------\left(i\right)$

The cube is cut into eight small cubes of equal volume.

Let the side of the which is cut into $8$ parts be $=x$

The volume of a small cube $=x^3$

The surface area of a small cube $=6x^2-------\left(ii\right)$

Volume of each small cube$$\begin{gathered} =\frac{1728}{8}c{m}^3 \\ =216c{m}^3 \end{gathered}$$

Or$$\begin{gathered} \Rightarrow x^3=216 \\ \Rightarrow x=6cm \end{gathered}$$

Or x = 6 cm

Surface areas of the cubes ratios$=$ $\frac{Surfaceareaofthebiggercube}{Surfaceareaofsmallercubes}$

From eq $\left(i\right)$ and $\left(ii\right),$, we get

Surface areas of the cubes ratios

$$\begin{gathered} =\frac{6a^2}{6x^2} \\ =\frac{a^2}{x^2} \\ =\frac{{12}^2}{6^2} \\ =\frac{4}{1} \end{gathered}$$

Therefore, the required ratio is $4:1$