Q : 8    A solid cube of side 12 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.

H Harsh Kankaria

This is an important question.

Given,

Side of a solid cube = $l = 12\ cm$

We know, the volume of a cube of side $l$ = $l^3$

$\therefore$ The volume of the given cube =$12^3\ cm^3$

Now, the cube is cut into 8 equal cubes of side $a$ (let)

$\therefore$ The total volume of these 8 cubes = Volume of the bigger cube

$\Rightarrow (8\times a^3)\ cm^3 = 12\ cm^3$

$\\ \Rightarrow a^3 = \frac{12}{8} \\ \Rightarrow a^3 = \left (\frac{12}{2} \right ) \\ \Rightarrow a = 6\ cm$

Therefore, the side of the new cube is $6\ cm$

Now, we know,

The surface area of a cube of side $l$ = $6l^2$

$\therefore$ The ratio between their surface areas $= \frac{Surface\ area\ of\ bigger\ cube}{Surface\ area\ of\ a\ smaller\ cube}$

$\\ = \frac{6l^2}{6a^2} \\ = \left (\frac{l}{a} \right )^2 \\ = \left (\frac{12}{6} \right )^2 \\ = 4$

Therefore, the ratio of their surface areas is $4:1$

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