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

Solution:

(b) 16 : 9
Let the radii of the spheres be R and r.
Then, ratio of their volumes

$=\frac{\frac{4}{3} \pi R^{3}}{\frac{4}{3} \pi r^{3}}$

Therefore,

$\frac{\frac{4}{3} \pi R^{3}}{\frac{4}{3} \pi r^{3}}=\frac{64}{27}$

$\Rightarrow \frac{R^{3}}{r^{3}}=\frac{64}{27}$

$\Rightarrow\left(\frac{R}{r}\right)^{3}=\left(\frac{4}{3}\right)^{3}$

$\Rightarrow \frac{R}{r}=\frac{4}{3}$

Hence, the ratio of their surface areas $=\frac{4 \pi R^{2}}{4 \pi r^{2}}$

$=\frac{R^{2}}{r^{2}}$

$=\left(\frac{R}{r}\right)^{2}$

$=\left(\frac{4}{3}\right)^{2}$

$=\frac{16}{9}$

 

$=16: 9$