The volume of one sphere is 27 times that of another sphere. Calculate the ratio of their:

(i) radii

(ii) surface areas

Given,

Volume of first sphere = 27 x volume of second sphere

Let the radius of the first sphere = r₁ and, radius of second sphere = r₂

(i) According to the question, we have :

$\Rightarrow \dfrac{4}{3} πr$1^3 = 27 \times \dfrac{4}{3} πr2^3 \\[1em] \Rightarrow r1^3 = 27 \times r2^3 \\[1em] \Rightarrow \dfrac{r1^3}{r2^3} = \dfrac{27}{1} \\[1em] \Rightarrow \Big(\dfrac{r1}{r2}\Big)^3 = \Big(\dfrac{3}{1}\Big)^3 \\[1em] \Rightarrow \dfrac{r1}{r2} = \dfrac{3}{1}.

Hence, r₁ : r₂ = 3 : 1.

(ii) Surface area of the first sphere = 4π(r₁)²

Surface area of second sphere = 4π(r₂)²

$\text{Ratio of surface areas} = \dfrac{4πr$1^2}{4πr2^2} \\[1em] = \dfrac{r1^2}{r2^2} = \Big(\dfrac{3}{1}\Big)^2 \\[1em] = \dfrac{9}{1}.

Hence, the ratio of surface areas = 9 : 1.