Question

The Surface areas of 2 Spheres are in the ratio 16:9. What is
the ratio of their volumes?​

Answer 1

Answer:

if the surface areas of two spheres are in the ratio 9 : 16, the ratio of their volumes is. 16 : 9. 27 : 64.

Answer 2

${\huge{\star{\underbrace{\tt{\red{Answer}}}}}}{\huge{\star}}$

$\Large{\blue{\tt{\underline{\underline{Given \ :-}}}}}$

${\sf{☞ \ Ratio \ of \ surface \ area \ of \ spheres \ = \ 16:9}}$

$\Large{\green{\tt{\underline{\underline{To \ find \ :-}}}}}$

• Ratio of volumes

$\Large{\orange{\tt{\underline{\underline{Solution \ :-}}}}}$

${\tt{Let \ radius \ of \ 1st \ sphere \ be \ r_{1} \ and \ the }}\\ {\sf{radius \ of \ 2nd \ sphere \ be \ r_{2} }}$

${\sf{✿ \ Ratio \ of \ surface \ area \ = \ 16:9}}$

${\boxed{\pink{\sf \ Surface \ area \ of \ sphere \ = \ 4\pi r²}}}$

${\sf{ \leadsto \ \frac{4\pi {r }^{2}_{1}}{4\pi {r}^{2}_{2} } = \frac{16}{9}}}$

${\sf{ \leadsto \ \frac{\cancel{4\pi } \: {r }^{2}_{1}}{\cancel{4\pi} \: {r}^{2}_{2} } = \frac{16}{9}}}$

${\sf{ \leadsto \ \frac{{r }^{2}_{1}}{{r}^{2}_{2} } = \frac{16}{9}}}$

${\sf{ \leadsto \ \frac{r _{1}}{r _{2} } = \sqrt{ \frac{16}{9} } }}$

${\sf{ \leadsto \ \frac{r _{1} }{r _{2}} = \frac{4}{3}}}$

Now

${\boxed{\purple{\sf \ Volume \ of \ sphere \ = \ \frac{4}{3} \pi r³}}}$

${\sf{\longmapsto \ Ratio \: of \: volumes \: = \frac{ \frac{4}{3}\pi {r_{1} }^{3} }{ \frac{4}{3} \pi {r _{2}}^{3} } }}$

${\sf{\longmapsto \ Ratio \: of \: volumes \: = \frac{ \cancel{\frac{4}{3}\pi }{r_{1} }^{3} }{\cancel{ \frac{4}{3} \pi } {r _{2}}^{3} }}}$

${\sf{\longmapsto \ Ratio \: of \: volumes \: = \frac{r _{1} ^{3} }{ {r _{2} }^{3} } }}$

${\sf{\Large{Substituting \ value \ of \ \frac{r_{1}}{r_{2}}}}}$

${\sf{\longmapsto \ Ratio \: of \: volumes \: = \frac{ {4}^{3} }{ {3}^{3} }}}$

${\sf{\longmapsto \ Ratio \: of \: volumes \: = \frac{64}{27} = 64 \ratio \: 27}}$

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$\Large\mathcal{\fcolorbox{lime}{black}{\red{Hope it's help ⚓⚓}}}$

$\large\mathcal{\fcolorbox{yellow}{teal}{\orange{Please mark me as brainliest (. ❛ ᴗ ❛.)}}}$