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.

⠀⠀⠀⠀━━━━━━━━━━━━━━━━━━━━━⠀⠀⠀⠀⠀

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

• 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,

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

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

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

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

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

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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,

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

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

Answer 2

${\huge{\rm{\underline{\underline{\rm{Question:-}}}}}}$

→ A solid cube of side 12 cm is cut into eight cubes of

equal volume. What will be the side of the new cube?

Also, find the ratio between their surface areas.

${\huge{\rm{\underline{\underline{Given:- }}}}}$

→  Side of Cube = 12cm  is divided into 8 cubes of equal volume.

${\huge{\rm{\underline{\underline{ To Find :-}}}}}$

→ The side of the new cube.

→ The ratio between their surface areas.

${\huge{\rm{\underline{\underline{Solution:-}}}}}$

⇒ Volume of cube =12 × 12 × 12 = 1728cm³

 

⇒ Volume of smaller cube

   

 =  $\frac{1}{8} \: of \: 1728$

 = $\frac{1}{8} \times 1728 = 216 \: cm ^{3}$

⇒  Let side be S.

  ∴ ( S )³   = 216

  ⇒ S =  6 cm

⇒ Ratio of their surface areas

 

   $= \frac{Surface \: area \: of \: bigger \: cube}{Surface \: area \: of \: smaller \: cube}$

   $= (\frac{12}{6} )^{2} = \frac{2^{2}} {1} = \frac{4}{1}$

∴ Ratio between their surface areas 4:1