Volume of a sphere varies directly with cube of length of its radius and surface area of sphere varies directly with the square of the length of radius. Prove that the square of volume of sphere varies directly with cube of its surface area.

Explanation

Let volume of sphere =V , area of surface =A and Radius =R .
As per question , V `prop R^3`
`rArr v =k_1 R^3 (when k_1 ne 0 =` variation constant ).
`therefore V =k_1R^3 ` …………(1)
Also A `prop R^2 rArr A =k_2R^2 ……….(2) (when k_2 ne 0= ` variation constant )
Now , from (1) we get `R^3 =(V) /(k_1) or , R =3sqrt(V/k_1) `
`therefore` from (2) we get , `A^3 =k_2^3 .(V^2)/(k_1^2) or , V^2 =(k_1^2) /(k_2^3). A^3`
or , `V ^2= k.A^3 [" when " k (k_1^2)/k_2^3 ne0, because k_1,k_2 ne 0]`
`rArr V^2 prop A^3 [ because k = ` non -zero variation constant ]
Hence the square of volume of sphere varies directly with cube of its surface area. (Proved )