Question

A cube and a sphere have equal surface areas. the ratio of their volumes is

Answer 1

Ah ! 

Solution : 

[tex]T.S.A. \ of \ a \ cube = 6a^2 \\ T.S.A. \ of \ a \ Sphere = 4 \pi r^2 \\ \\ [/tex]

ATQ -> 
[tex]6a^2 = 4 \pi r^2 \\ =\ \textgreater \ ( \frac{a}{r} )^2 = ( \frac{4 \pi }{6} ) \\ \\ =\ \textgreater \ ( \frac{a}{r} )^3 = ( \frac{4 \pi }{6} )^{ \frac{3}{2} } \\ \\ =\ \textgreater \ ( \frac{a}{r} )^3 ( \frac{3}{4 \pi } ) = ( \frac{3}{4 \pi } ) ( \frac{4 \pi }{6} )^{ \frac{3}{2} } = \sqrt{( \frac{ \pi }{6} )} = 0.12 ( approx. ) \\ \\ =\ \textgreater \ \frac{Volume \ of \ cube}{Volume \ of \ sphere} = \frac{3}{25} [/tex]

Hence, the ratio of their VOLUMES = [ 3 : 25 ] ( approx. ) 
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The exact answer = $\sqrt{\pi} : \sqrt{6}$