The capacities of two hemispherical vessels are 6.4 litres and 21.6 litres. The areas of inner curved surfaces of the vessels will be in the ratio of :

A. $$\sqrt 2 $$ : $$\sqrt 3 $$

B. 2 : 3

C. 4 : 9

D. 16 : 81

Answer: Option C

Solution(By Examveda Team)

Let their radii be R and r
Then,
$$\eqalign{ & \frac{{\frac{2}{3}\pi {R^3}}}{{\frac{2}{3}\pi {r^3}}} = \frac{{6.4}}{{21.6}} \cr & \Rightarrow {\left( {\frac{R}{r}} \right)^3} = \frac{8}{{27}} \cr & \Rightarrow {\left( {\frac{R}{r}} \right)^3} = {\left( {\frac{2}{3}} \right)^3} \cr & \Rightarrow \frac{R}{r} = \frac{2}{3} \cr} $$
∴ Ratio of curved surface area :
$$ = \frac{{2\pi {R^2}}}{{2\pi {r^2}}} = {\left( {\frac{R}{r}} \right)^2} = \frac{4}{9}\,or\,4:9$$