# Choose the correct answer of the following question:

Question:

Choose the correct answer of the following question:

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

(a) 64 : 27                          (b) 16 : 9                          (c) 4 : 3                          (d) 163 : 93

Solution:

Let the radius of the two spheres be $r$ and $R$.

As,

$\frac{\text { Surface area of the first sphere }}{\text { Surface area of the second sphere }}=\frac{16}{9}$

$\Rightarrow \frac{4 \pi R^{2}}{4 \pi r^{2}}=\frac{16}{9}$

$\Rightarrow\left(\frac{R}{r}\right)^{2}=\frac{16}{9}$

$\Rightarrow \frac{R}{r}=\sqrt{\frac{16}{9}}$

$\Rightarrow \frac{R}{r}=\frac{4}{3} \quad \ldots .(\mathrm{i})$

Now,

The ratio of their volumes $=\frac{\text { Volume of the first sphere }}{\text { Volume of the second sphere }}$

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

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

$=\left(\frac{4}{3}\right)^{3} \quad[\operatorname{Using}(\mathrm{i})]$

$=\frac{64}{27}$

$=64: 27$

Hence, the correct answer is option (a).