Question

the volume of two spheres are in the ratio 64:27,then the ratio of their surface areas is​

Answer 1

$\Huge{\mathfrak{\red{\underline{\green{Solution}}}}}$

Given:-

• Ratio of volume of spheres = 64 :27.

To Find:-

• Ratio of their surface areas = ?

Solution:-

Let's understand

Here, ratio of volume of spheres is 64:27 and we have to find ratio of their surface areas.

We know that

$\large{\sf{\fbox{\red{Volume\:of\:sphere={\frac{4}{3}π{r}^{3}}}}}}$

Let assume:

• Let $\sf{r_1}$=Radius of first sphere
• Let$\sf{r_2}$=Radius of second sphere

According to the question

$\sf \implies \: \frac{volume \: of \: first \: sphere}{volume \: of \: second \: sphere} = \frac{64}{27} \\ \\ \sf \implies \: \frac{ \frac{4}{3} \pi \: {r1}^{3} }{ \frac{4}{3} \pi \: {r2}^{3} } = \frac{64}{27} \\ \\ \sf \implies \: \frac{ {r1}^{3} }{ {r2}^{3} } = \frac{64}{27} \\ \\ \sf \implies \: {( \frac{r1}{r2} )}^{3} = \frac{64}{27} \\ \\ \sf \implies \: \frac{r1}{r2} = \sqrt[3]{ \frac{64}{27} } = \frac{4}{3}$

Hence,r1/r2 = 4/3.

We know that,

$\large{\sf{\fbox{\green{Surface\:area\:of\: sphere=4π{r}^{2}}}}}$

• where π = 22/7
• r = Radius of a sphere

Now,

$\sf \implies \: \frac{surface \: area \: of \: first \: sphere}{surface \: area \: of \: second \: sphere} = \frac{4 \pi \: {r1}^{2} }{4 \pi \: {r2}^{2} } = \frac{ {r1}^{2} }{ {r2}^{2} } = ( { \frac{r1}{r2} }^{2} ) \\ \\ \\ \sf \implies \: \frac{surface \: area \: of \: first \: sphere}{surface \: area \: of \: second \: sphere} = ( { \frac{4}{3} }^{2} ) = \frac{16}{9}$

Therefore, ratio of surface area of two sphere is 16 :9.