Question

12. 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.
( FULL EXPLANATION)​

Answer 1

☯ 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.

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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,

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

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

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

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

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

⠀⠀⠀⠀━━━━━━━━━━━━━━━━━━━━━⠀⠀⠀⠀⠀

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

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

⋆ Surface area of cube is given by,

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

Therefore,

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

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

Answer 2

A chemical equation is the symbolic representation of a chemical reaction in the form of symbols and formulae, wherein the reactant entities are given on the left-hand side and the product entities on the right-hand side. ... The first chemical equation was diagrammed by Jean Beguin in 1615.