Question

A solid cube of side 12 cm is cut into eight cubes of
equal volume. What will be the side of the new cube? Find the ratio between their surface areas.​

Answer 1

Answer:

Answer

Given:

Side 12cm

⇒Volume of cube =12×12×12=1728cm3

⇒Volume of smaller cube 

=81 of 1728

=81×1728=216cm3

⇒Let side be S.

∴ (S)3=216

⇒S=6 cm

Ratio of SA of cubes 

=SA of smaller cubeSA of bigger cube

=(612)2=122=14

∴ Ratio between their surface areas 4:1

Answer 2

★Let's consider side of new cube be a cm.

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$\underline{\bigstar\:\boldsymbol{According\:to\:the\:question\::-}}$

Solid cube of side 12 cm is divided into 8 cubes of equal volume.

⠀⠀⠀

Therefore,

$:\implies\sf Volume\:of\:big\:cube = 8 \times Volume\:of\:small\:cube$

$\dag\;{\underline{\frak{As\;we\;know\;that,}}}$

⋆ Volume of cube is given by,

$\star\;{\boxed{\sf{\purple{Volume_{\;(cube)} = (side)^3}}}}$

$\sf Here \begin{cases} & \sf{Side\:of\:big\:cube = \bf{12\:cm}} \\ & \sf{Side\:of\:small\:cube = \bf{a\:cm}} \end{cases}$

$\dag\;{\underline{\frak{Now,\:Putting\:values,}}}$

$:\implies\sf (12)^3 = 8 \times (a)^3\\ \\ \\ :\implies\sf 12 \times 12 \times 12 = 8 \times (a)^3\\ \\ \\ :\implies\sf \dfrac{1}{ \cancel{8}} \times \cancel{12} \times 12 \times 12 = a^3\\ \\ \\ :\implies\sf \dfrac{1}{\cancel{4}} \times 6 \times \cancel{12} \times 12 = a^3\\ \\ \\ :\implies\sf \dfrac{1}{ \cancel{2}} \times 6 \times 6 \times \cancel{12} = a^3\\ \\ \\ :\implies\sf 6\times 6 \times 6 = a^3\\ \\ \\ :\implies\sf a^3 = (6)^3\\ \\ \\ :\implies\sf \sqrt[3]{a^3} = \sqrt[3]{6^3}\\ \\ \\ :\implies{\underline{\boxed{\frak{\pink{a = 6}}}}}\;\bigstar$

$\therefore\:{\underline{\sf{Side\:of\:small\:or\:new\:cubes\:is\: {\textbf{\textsf{6\:cm}}}.}}}$

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$\dag\;{\underline{\frak{Finding\:ratio\:of\:their\:surface\:areas,}}}$

$\dag\;{\underline{\frak{As\;we\;know\;that,}}}$

⋆ Surface area of cube is given by,

$\star\;{\boxed{\sf{\pink{TSA_{\;(cube)} = 6 \times (side)^2}}}}$

Therefore,

$:\implies\sf \dfrac{Surface\:area_{\:(big\:cube)}}{Surface\:area_{\:(small\:cube)}}\\ \\ \\ :\implies\sf \dfrac{6 \times (12)^2}{6 \times (6)^2}\\ \\ \\ :\implies\sf \dfrac{\cancel{6 \times 12 \times 12}}{\cancel{6 \times 6 \times 6}}\\ \\ \\ :\implies\sf \dfrac{4}{1}\\ \\ \\ :\implies{\underline{\boxed{\frak{\green{4:1}}}}}\;\bigstar$

$\therefore\:{\underline{\sf{Thus,\:the\:ratio\:their\:surface\:area\:is\: {\textbf{\textsf{4:1}}}.}}}$

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We all Done :D