Question

Question 20
Volumes of two spheres are in the ratio 64 : 27 , then the ratio of their surface area is
(A) 3 : 4
(B) 4 : 3
(C) 9 : 16
(D) 16 : 9

Answer 1

Option (D) is correct.
Let the radii of the two spheres are $r_{1}and{r}_2$ respectively.
$\therefore$ Volume of the sphere of radius
$r_1=v_1=\frac{4}{3}\pi r_1^3\dots \left(i\right)$
$[\because Volumeofsphere=\frac{4}{3}\pi (radius{)}^3]$

Volume of the sphere of radius
$r_2=v_2=\frac{4}{3}\pi r_2^3\dots .\left(ii\right)$

Given, ratio of volumes
$=v_1:v_2=64:27\Rightarrow \frac{\frac{4}{3}\pi r_1^3}{\frac{4}{3}\pi r_2^3}=\frac{64}{27}$ [ using Equations (i) and (ii)]

$\Rightarrow \frac{r_1^3}{r_2^3}=\frac{64}{27}\Rightarrow \frac{r_1}{r_2}=\frac{4}{3}...\left(iii\right)$

Now, ratio of surface area
$=\frac{4\pi r_1^2}{4\pi r_2^2}[\because surfaceareaofsphere=4\pi (radius{)}^2]$

$=\frac{r_1^2}{r_2^2}$

$={\left(\frac{r_1}{r_2}\right)}^2={\left(\frac{4}{3}\right)}^2[UsingEq.(iii\left)\right]$

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