Question

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

Answer 1

Given

• Ratio of volume of two spheres is 64:27

To Find

• Ratio of their surface areas ?

Solution

We know that,

$\boxed{ \tt{Volume \: of \: sphere = \frac{4}{3}\pi {r}^{3} }}$

let ‘v1’ and ‘v2’ are volume of two spheres and ‘r1’ and ‘r2’ are radius of these spheres respectively.

$\implies \bold{v1 = \dfrac{4}{3}\pi {r1}^{3} }$

$\implies { \bold{v2 = \dfrac{4}{3}\pi {r2}^{3} }}$

Now, ratio of v1 and v2

$\implies \bold{v1 : v2 = \dfrac{4}{3} \pi {r1}^{3} \div \dfrac{4}{3}\pi {r2}^{3} } \\ \\ \implies \bold{ \dfrac{64}{27} = \cancel {\dfrac{4}{3} \pi} \: {r1}^{3} \div \cancel{\dfrac{4}{3} \pi} \: {r2}^{3} } \: \: \: \: \: \: \\ \\ \implies \bold{ \dfrac{64}{27} = \dfrac{ {r1}^{3} }{ {r2}^{3} } } \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \\ \\ \implies \bold{ \dfrac{ {4}^{3} }{ {3}^{3} } = \dfrac{ {r1}^{3} }{ {r2}^{3} }} \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \\ \\ \implies \bold{ \dfrac{r1}{r2} = \dfrac{4}{3} \: \: \: \: ....(i) } \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \:$

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Now, we know that

$\boxed{ \bold{surface \: area \: of \: sphere = 4\pi {r}^{2} }}$

let ‘s1’ and ‘s2’ are surface areas of two spheres

$\implies \bold{s1 = 4 {r1}^{2} } \\ \\ \implies \bold{s2 = 4 {r2}^{2} }$

now we get,

$\bold{\frac{s1}{s2} = \cancel \frac{4 {r1}^{2} }{4 {r2}^{2} } } \\ \\ \bold{ \implies \dfrac{s1}{s2} = \dfrac{ {r1}^{2} }{ {r2}^{2} } }$

Now, put the value of r1/r2 from (i)

$\implies \bold{ \dfrac{s1}{s2} = \dfrac{ {4}^{2} }{ {3}^{2} } } \: \: \: \: \: \: \: \: \: \: \: \: \: \\ \\ \bold{ \implies \dfrac{s1}{s2} = \dfrac{16}{9} } \: \: \: \: \: \: \: \: \: \: \: \: \: \\ \\ \implies \bold{s1 :s2 = 16 :9 }$

Hence, ratio of the surface area of two spheres is 16:9