Question

4
Volumes of two spheres are in the ratio 64:27. The ratio of their surface areas is---​

Answer 1

Given data :-

• Volumes of two spheres are in the ratio 64:27.

Solution:-

Let, first spheres radius be R and second spheres radius be r

→ Volumes first spheres : volume of second spehere = 64:27

Here, we use formula of volume of sphere

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

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

${ \bf \:{ {( \frac{R}{r} )}^{3} = \frac{64}{27} }}$

${ \bf \:{ {\frac{ \: R }{ \:r } } = \sqrt[3]{ \frac{64}{27} } }}$

${ \bf \:{ {\frac{ \: R }{ \: r } } = \frac{4}{3} }}$

Now, we use formula of surface area of sphere

${ \bf \:{ \frac{surface \: area \: of \: first \: sphere}{surface \: area \: of \: second \: sphere} = \frac{4 \: \pi \: {R}^{2} }{4 \: \pi \: {r}^{2} } }}$

${ \bf \:{ \frac{surface \: area \: of \: first \: sphere}{surface \: area \: of \: second \: sphere} = \frac{ \: {R}^{2} }{ \: {r}^{2} } }}$

Now, put value of R and r in equation

${ \bf \:{ \frac{surface \: area \: of \: first \: sphere}{surface \: area \: of \: second \: sphere} = \frac{16}{ 9 } }}$

Hence, the ratio of the surface areas of the two sphere is 16:9

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