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Question
The surface areas of a sphere and a cube are equal. Find the ratio of their volumes.
Solution
Surface area of the sphere = 4πr2
Surface area of the cube= 6a2
Therefore,
4πr2 = 6a2
⇒ 2πr2 = 3a2
`=>2pi"r"^2 = 3"a"^2`
`=> r^2 = (3"a"^2)/(2pi)`
`=>"r" = sqrt(3/(2pi)a)`
Ratio of their Volumes `= (4//3 pi"r"^3)/a^3 = (4pi"r"^3)/3"a"^3`
`= (4pi)/"3a"^3xx(3sqrt(3"a"^3))/(2pisqrt(2pi)) ["since" "r" = sqrt(3/2pi)]`
`=(2sqrt(3))/sqrt(2pi)`
`=(2sqrt(3))/(sqrt(2)xxsqrt(22)/sqrt(7)`
`= (2xxsqrt(3)xxsqrt(7))/(sqrt(2)xxsqrt(2)xxsqrt(11))`
`=sqrt(21)/sqrt(11)`
Thus, the ratio of their volumes is `sqrt(21):sqrt(11)`
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