What is the ratio of the volume of cube and the smallest sphere that could contain it?
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You need to fit the sphere inside the cube, hence, the diameter of sphere needs to be equal to the side of cube such that:
`D = a`
D represents the diameter of the sphere
a represents the side of the cube
You need to remember the formula of volume of sphere such that:
`V = (4pir^3)/3`
You need to remember that `r = D/2 =gt r^3 = (D^3)/8`
Substituting `(D^3)/8` for`r^3` in formula of volume of sphere yields:
`V = (4pi*(D^3)/8)/3`
`V = (pi*(D^3)/2)/3 =gt V = (pi*(D^3))/6`
Notice that D = a, hence, you need to substitute a for D in formula of volume of sphere such that:
`V = (pi*(a^3))/6`
You need to remember the formula of volume of the cube such that:
`v = a^3`
You need to evaluate the ratio `v/V` such that:
`v/V = (a^3)/((pi*(a^3))/6)`
Reducing by `a^3` yields:
`v/V = 6/pi`
Hence, evaluating the ratio of volumes under given conditions yields `v/V = 6/pi.`
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