Question

Volumes of two spheres are in the ratio 64 : 27. The ratio of their surface areas is ______.

Options

• 3 : 4

• 4 : 3

• 9 : 16

• 16 : 9

Answer 1

Volumes of two spheres are in the ratio 64 : 27. The ratio of their surface areas is 16 : 9.

Explanation:

Let the radii of the two spheres are r₁ and r₂, respectively.

∴ Volume of the sphere of radius,

r₁ = V₁ = `4/3 pi"r"_1^3`   ...(i) [∵ Volume of sphere = `4/3pi` (radius)³]  

And volume of the sphere of radius,

r₂ = V₂ = `4/3 pi"r"_2^3`    ...(ii)

Given, ratio of volumes = V₁ : V₂ = 64 : 27

⇒ `(4/3 pi"r"_1^3)/(4/3 pi"r"_2^3) = 64/27`  ...[Using equations (i) and (ii)]

⇒ `("r"_1^3)/("r"_2^3) = 64/27` 

⇒ `"r"_1/"r"_2 = 4/3`   ...(iii)

Now, ratio of surface area = `(4 pi"r"_1^2)/(4 pi"r"_2^2)`   ...[∵ Surface area of a sphere = 4π (radius)²]

= `"r"_1^2/"r"_2^2`

= `("r"_1/"r"_2)^2`

= `(4/3)^2`   ...[Using equation (iii)]

= 16 : 9 

Hence, the required ratio of their surface area is 16 : 9.