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.

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Solution

Step I: Finding side of the new cubeSide of the big solid cube $=12cm$We know volume of a cube $=Side×Side×Side$So, Volume of the solid cube $=12×12×12=1728c{m}^{3}$Given that the solid cube is cut into $8$ small cubes of equal volumes.Thus, Volume of each small cube $=\frac{1}{8}×1728=216c{m}^{3}$Let the side of small cube is $a.$So the volume of a small cube $=a×a×a={a}^{3}$Thus, equating the volumes of the small cube, we get:${a}^{3}=216$$a=6cm$Thus, side of the new cube is $6cm$.Step II: Finding the ratio of surface areas.We know, the Surface area of cube is given by $SA=6×sid{e}^{2}$Surface area of the big solid cube of side $12cm=6×{\left(12\right)}^{2}=864c{m}^{2}$Surface area of the small cube of side $6cm=6×{\left(6\right)}^{2}=216c{m}^{2}$Ratio of $SA$of the cubes $=$ $\frac{SAofsolidcube}{SAofsmallcube}$ $=\frac{864}{216}\phantom{\rule{0ex}{0ex}}=\frac{4}{1}\phantom{\rule{0ex}{0ex}}=4\mathbf{:}\mathbf{}\mathbf{1}$Thus, the ratio of surface areas of two cubes is $4\mathbf{:}\mathbf{1}$.

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