Math, asked by karthikkumar98, 11 months ago



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.​

Answers

Answered by renuagrawal393
13

Answer:

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Answered by PD626471
1009

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

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\begin{gathered}\underline{\bigstar\:\boldsymbol{According\:to\:the\:question\::}}\\ \\\end{gathered}

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

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Therefore,

\begin{gathered}:\implies\sf Volume\:of\:big\:cube = 8 \times Volume\:of\:small\:cube\\ \\\end{gathered}

\begin{gathered}\dag\;{\underline{\frak{As\;we\;know\;that,}}}\\ \\\end{gathered}

\sf{⋆ Volume  \: of  \: cube \:  is  \:  \: given \:  by,}

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

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

\begin{gathered}\dag\;{\underline{\frak{Now,\:Putting\:values,}}}\\ \\\end{gathered}

\begin{gathered}:\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\\ \\\end{gathered}

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

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\begin{gathered}\dag\;{\underline{\frak{Finding\:ratio\:of\:their\:surface\:areas,}}}\\ \\\end{gathered}

\begin{gathered}\dag\;{\underline{\frak{As\;we\;know\;that,}}}\\ \\\end{gathered}

\sf{⋆ Surface  \: area \:  of  \: cube  \: is  \: given  \: by,}

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

\sf{Therefore,}

\begin{gathered}:\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\\ \\\end{gathered}

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

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