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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$=864:216=4:1$.