Question

If the ratio of volumes of two spheres is 1 : 8, then the ratio of their surface areas is 

Options

• 1 : 2

• 1 : 4

• 1 : 8

• 1 : 16

Answer 1

Here, we are given that the ratio of the two spheres of ratio 1:8

Let us take,

The radius of 1^st sphere = r₁

The radius of 1^st sphere = r₂

So,

Volume of 1^st sphere (V₁) =  `4/3 pi r_1^3`

Volume of 2^nd sphere (V₂) = `4/3 pi r_2^3`

Now,  `V_1/V_2 = 1/8`

`((4/3 pi r_1^3))/((4/3 pi r_2^1)) = 1/8`

           `r_1^3/r_2^3 = 1/8`  

          `r_1/r_2 = 3sqrt(1/8)`

          `r_1/r_2 = 1/2`                .............(1)

Now, let us find the surface areas of the two spheres

Surface area of 1^st sphere (S₁) =  `4 pi r_1^2`

Surface area of 2^nd sphere (S₂) = `4 pi r_2^2`

So, Ratio of the surface areas,

`S_1/S_2 = (4pir_1^2)/(4 pi r_2^2)`

        `=r_1^2/r_2^2`

       ` = (r_1/r_2)^2`

Using (1), we get,

`S_1 /S_2 = ( r_1/r_2)^2`

          `= (1/2)^2`

         `=(1/4)`

Therefore, the ratio of the spheres is 1 : 4 .