If volume of two spheres are in the ratio `64: 27`, then the ratio of their surface area is

Options

A. `3:4`
B. `4:3`
C. `9:16`
D. `16:9`

Answer

D

Explanation

Let the radii of two spheres are `r_(1) ` and `r_(2)` respectively .
`therefore ` Volume of the sphere of radius , `r_(1) = V_(1) = (4)/(3) pir_(1)^(3)" "……..(i) " "[therefore "Volume of sphere "= (4)/(3) pi ("radius")^(3)]`
and volume of the sphere of radius , `r_(2) = v_(1) = (4)/( 3) pi r_(2)^(3)" "....(ii)`
Given, ratio of volume `= V_(1): V_(2)= (64)/(27) rArr ((3)/(4) pir_(1)^(3))/((4)/(3)pir_(2)^(3))" "["Using Eqs, (i) and (ii)"]`
`rArr " "(r_(1)^(3))/(r_(2)^(3)) =(64)/(27)rArr (r_(1))/(r_(2)) = (4)/(3)" "......(i)`
Now, ratio of surface area `= (4pir_(1)^(2))/(4pir_(2))" "[therefore "Surface area of a sphere"= 4 pi ("radius")^(2)]`
`" "(r_(1)^(2))/(r_(2)^(2))`
`" "= ((r_(1))/(r_(2)))^(2) = ((4)/(3))^(2) " "["Using Eq. (iii)"]`
Hence, the required ratio of their surface area is `16:9`