The volume of the two spheres are in the ratio 64:27. Find the difference of their surface areas, if the sum of their radii is 7.

Explanation

Let the radius of two spheres be `r_(1)` and `r_(2)`
Given, the ratio of the volume of two spheres = 64: 27
`(V_(1))/(V_(2)) =(64)/(27) rArr ((4)/(3)pir_(1)^(3))/((4)/(3)pir_(2)^(3)) = (64)/(27)`
`rArr" "((r_(1))/(r_(2)))^(3) = ((4)/(3))^(3) " "[because "volume of sphere" =(4)/(3) pir^(3)]`
`rArr " "(r_(1))/(r_(2)) =(4)/(3)`
Let the surface areas of the two spheres `S_(1)` and `S_(2)`
`therefore" "(S_(1))/(S_(2)) = (4pir_(1)^(2))/( 4pir_(2)^(2)) = ((r_(1))/(r_(2)))^(2) rArr S_(1),S_(2) = ((4)/(3))^(2) = (16)/(9)`
`rArr" "S_(1),S_(2) = 16:9`
Hence, the ratio of the their surface areas is 16: 9.