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

## We know volume of cube $={\left(side\right)}^{3}$ $={\left(12\right)}^{3}$ $=1728{m}^{3}$Surface area of cube$=6{a}^{2}$ $=6×{\left(12\right)}^{2}-------------\left(i\right)$The cube is cut into eight small cubes of equal volume.Let the side of the which is cut into $8$ parts be $=x$The volume of a small cube $={x}^{3}$The surface area of a small cube $=6{x}^{2}-------\left(ii\right)$Volume of each small cube$=\frac{1728}{8}c{m}^{3}\phantom{\rule{0ex}{0ex}}=216c{m}^{3}$Or$⇒{x}^{3}=216\phantom{\rule{0ex}{0ex}}⇒x=6cm$Or x = 6 cmSurface areas of the cubes ratios$=$ $\frac{Surfaceareaofthebiggercube}{Surfaceareaofsmallercubes}$From eq $\left(i\right)$ and $\left(ii\right),$, we getSurface areas of the cubes ratios$=\frac{6{a}^{2}}{6{x}^{2}}\phantom{\rule{0ex}{0ex}}=\frac{{a}^{2}}{{x}^{2}}\phantom{\rule{0ex}{0ex}}=\frac{{12}^{2}}{{6}^{2}}\phantom{\rule{0ex}{0ex}}=\frac{4}{1}$Therefore, the required ratio is $4:1$

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