A $4\ cm$ cube is cut into $1\ cm$ cubes. Find the total surface area of all small cubes. What is the ratio of the surface area of the smaller cube to that of the larger cube?

Given:

A $4\ cm$ cube is cut into $1\ cm$ cubes.

To do:

We have to find the total surface area of all the small cubes and the ratio of the surface area of the smaller cube to that of the larger cube.

Solution:

Volume of $4\ cm$ cube$=(4\ cm)^3=64\ cm^3$

Volume of $1\ cm$ cube$=(1\ cm)^3=1\ cm^3$

Total number of $1\ cm$ cubes$=\frac{Volume\ of\ 4\ cm\ cube}{Volume\ of\ 1\ cm\ cube}$

$=\frac{64}{1}=64$

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

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

Total surface area of $64$ small cube$=64\times6\ cm^2=384\ cm^2$

Total surface area of the large cube$=6(4\ cm)^2=6\times16\ cm^2=96\ cm^2$

The ratio of the surface area of the smaller cube to that of the larger cube$=6:96=1:16$.