If the curved surface areas of a sphere and a right circular cylinder both of same radii be equal , then what will be the ratio of their volumes ?

Explanation

Let the radii of both the sphere and the cylinder be r unit also let the height of the cylinder be h unit . ` :. ` the curved surface area of the sphere ` = 4pir^(2) ` sq - unit
and the curved surface area of the cylinder = ` 2pirh ` sq - unit
As per question , ` 4 pir^(2) = 2pirh`
or, `(pir^(2))/(pirh) = 2/4` or , `r/h = 1/2 or, h = 2r`
`therefore` the ratio of the volumes of the sphere and the cylinder
` = 4/3 pi r^(3) : pi r^(3) h `
` = (4/3 pir^(3))/(pir^(2)h) = (4pir^(3))/(3pir^(2) xx2r) [ :' h = 2r ] `
` = (4pir^(3))/(6pir^(3)) = 4/6 = 2/3 = 2 :3 `
Hence the required ratio of the volumes of sphere and the cylinder = 2 : 3