Question

Find the ratio of surface areas of cubes whose volumes are 216^3 and 512^3respectively​

Answer 1

Answer:

The ratio of surface areas of two cubes whose volumes are 216 cm³ and 512 cm² is 9 : 16.

Step-by-step explanation:

${\underline{\underline{\bf{Given:}}}}$

• $\sf{Volume\:of\:cube\:_1=216\:cm^3}$
• $\sf{Volume\:of\:cube\:_2=512\:cm^3}$

${\underline{\underline{\bf{Need\:to\:find:}}}}$

• $\textsf{The ratio of TSA of the cubes}$

${\underline{\underline{\bf{Formulae\:to\:be\:used:}}}}$

$\underline{\boxed{\sf{{Volume}_{[Cube]}=a^3\:cu.units}}}$

$\underline{\boxed{\sf{{TSA}_{[Cube]}=6a^2\:sq.units}}}$

$\:\:\:\:\:\:\:\:\:\sf{\bullet\:a=side}$

${\underline{\underline{\bf{Solution:}}}}$

As the measures of side of the cubes aren't given, let's find them first by inserting the measure of volume of cubes in the formula:

$\leadsto{\sf{\purple{Volume\:(Cube)=a^3\:cu.units}}}$

${\boxed{\bf{Cube\:_1}}}$

$\:\:\:\:\:\::\implies{\sf{216=a^3}}$

$\:\:\:\:\:\:\:\:\:\::\implies{\sf{ \sqrt[3]{216}=a}}$

$\:\:\:\:\:\:\:\:\:\:\:\:\:\::\implies\underline{\bf{6\:cm=a}}$

${\boxed{\bf{Cube\:_2}}}$

$\:\:\:\:\:\::\implies{\sf{512=a^3}}$

$\:\:\:\:\:\:\:\:\:\::\implies{\sf{ \sqrt[3]{512}=a}}$

$\:\:\:\:\:\:\:\:\:\:\:\:\:\::\implies\underline{\bf{8\:cm=a}}$

We've obtained the measures of sides of the cubes. So, now let's find the TSA of the cubes by inserting the measure of side in the formula:

$\leadsto{\sf{\purple{TSA\:(Cube)=6a^2\:sq.units}}}$

${\boxed{\bf{Cube\:_1}}}$

$\:\:\:\:\:\::\implies{\sf{6\times 6^2}}$

$\:\:\:\:\:\:\:\:\:\::\implies{\sf{6\times 6\times 6}}$

$\:\:\:\:\:\:\:\:\:\:\:\:\:\::\implies{\sf{36\times 6}}$

$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\::\implies\underline{\bf{216\:cm^2}}$

${\boxed{\bf{Cube\:_2}}}$

$\:\:\:\:\:\::\implies{\sf{6\times 8^2}}$

$\:\:\:\:\:\:\:\:\:\::\implies{\sf{6\times 8\times 8}}$

$\:\:\:\:\:\:\:\:\:\:\:\:\:\::\implies{\sf{6\times 64}}$

$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\::\implies\underline{\bf{384\:cm^2}}$

Finally, the ratio of TSA of the cubes is:

$\rightarrowtail{\sf{\dfrac{216}{384}}}$

$\:\:\:\:\:\:\:\:\:\sf{=\dfrac{108}{192}}$

$\:\:\:\:\:\:\:\:\:\sf{=\dfrac{54}{96}}$

$\:\:\:\:\:\:\:\:\:\sf{=\dfrac{27}{48}}$

$\:\:\:\:\:\:\:\:\:\sf{=\dfrac{9}{16}}$

$\:\:\:\:\:\:\:\:\:\frak{=\boxed{\frak{\red{9:16}}}}$

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