Question

The ratio of the surface areas of two spheres is 9 :16, find the ratio of their:

1. radii 
2. volumes

Answer 1

Given:

Ratio of the surface areas of two spheres

S₁ : S₂ = 9 : 16

To Find:

1. r1:r2r 
2. V₁:V₂

(i) Ratio of radii

Surface area of sphere:

S = 4πr²

`S_1/S_2 = (4πr_1^2)/(4πr_2^2) = r_1^2/r_2^2`

Given `S_1/S_2 = 9/16` therefore:

`r_1^2/r_2^2 = 9/16`   ...[Taking square root]

`r_1/r_2 = 3/4`

r₁ : r₂ =  3 : 4

(ii) Ratio of volumes

Volume of sphere:

V = `4/3 πr^3`

so, 

`V_1/V_2 = r_1^3/r_2^3`

Using r₁ : r₂ = 3 : 4

V₁ ​: V₂ ​= 3³ : 4³ = 27 : 64

V₁ ​: V₂ ​= 27 : 64