A solid cube of side $ 12 \mathrm{~cm} $ 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.

Given:

A solid cube of side \( 12 \mathrm{~cm} \) is cut into eight cubes of equal volume. 

To do:

We have to find the side of the new cube and the ratio of the surface area of the larger cube to that of the smaller cube.

Solution:

Volume of $12\ cm$ cube$=(12\ cm)^3$

$=1728\ cm^3$

Volume of each small cube $=\frac{1728}{8}\ cm^3$

$=216\ cm^3$

Let the side of each small cube be $s$.

This implies,

Volume of each small cube $=(s\ cm)^3=216\ cm^3$

$s^3=(6)^3$

$s=6\ cm$

Total surface area of a cube of side $a$ is $6a^2$.

Therefore,

Total surface area of the large cube$=6(12\ cm)^2=6\times144\ cm^2=864\ cm^2$

Total surface area of $1$ small cube$=6(6\ cm)^2=216\ cm^2$

The ratio of the surface area of the larger cube to that of the smaller cube$=864:216=4:1$.