Question

The sphere and cube have same surface. Show that the ratio of the volume of sphere to that of cube is `root`6 : `root``pi`

Answer 1

Radius of given ball r = 1.5 cm
Radius of first ball `r_(1)=0.75 cm`
Radius of second ball `r_(2)=1 cm`
Let the radius of 3rd ball `= r_(3)`
Now,
volume of first ball + volume of second ball + volume of third ball = volume of given ball
`rArr (4)/(3)pi r_(1)^(3)+(4)/(3)pi r_(2)^(3)+(4)/(3)pi r_(3)^(3)=(4)/(3)pi r^(3)`
`rArr r_(1)^(3)+r_(2)^(3)+r_(3)^(3)=r^(3)`
`rArr r_(3)^(3)=r^(3)-r_(1)^(3)-r_(2)^(3)`
`= (1.5)^(3)-(0.75)^(3)-(1)^(3)`
`=((3)/(2))^(3)-((3)/(4))^(3)-(1)^(3)`
`=(27)/(8)-(27)/(64)-1=(216-27-64)/(64)`
`rArr r = (216-91)/(64)=(125)/(64)=((5)/(4))^(3)`
`=(1.25)^(3)`
`rArr = r_(3)= 1.25 cm`
`therefore` Radius of third ball = 1.25 cm