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

Answer : $16:9$

We have been given that the volume of two sphere is in the ratio $=64:27$

We know that, Volume of sphere is given as $\frac{4}{3}\pi r^{3}$ (where r is its radius)

And surface area is given as $4\pi r^{2}$

Let volume of sphere $1=V_{1}$         (radius $r_{1}$)

And, volume of sphere $2=V_{2}$        (radius $r_{2}$)

Then, $\frac{V_{1}}{V_{2}}=\frac{64}{27}$

$\frac{\frac{4}{3}\pi r_{1}^{3}}{\frac{4}{3}\pi r_{2}^{3}}=\frac{64}{27}$

$\frac{r_{1}^{3}}{r_{2}^{3}}=\left ( \frac{4}{3} \right )^{3}$

$\frac{r_{1}}{r_{2}}=\frac{4}{3}$

Then, ratio of areas of both spheres

$\frac{\text {area of sphere (1))}}{\text {area of sphere }(2)}=\frac{4\pi r_{1}^{2}}{4\pi r_{2}^{2}}$

$=\frac{r_{1}^{2}}{r_{2}^{2}}=\left ( \frac{r_{1}}{r_{2}} \right )^{2}=\left ( \frac{4}{3} \right )^{2}=\frac{16}{9}$

Hence the required ratio is $16:9$

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