picorants03p

2023-03-06

If the ratio of volumes of two spheres is 1 : 8 then the ratio of their surface areas is?

popescy91

Violume of first sphere $=\frac{4}{3}\pi {r}_{1}^{2}$
and volume of second sphere $=\frac{4}{3}\pi {r}_{2}^{3}$
But ratio in their volumes = 1 : 8
$\therefore \frac{\frac{4}{3}\pi {r}_{1}^{3}}{\frac{4}{3}\pi {r}_{2}^{3}}=\frac{1}{8}\phantom{\rule{0ex}{0ex}}⇒\frac{{r}_{1}^{3}}{{r}_{2}^{3}}=\frac{1}{8}=\left(\frac{1}{2}{\right)}^{3}\phantom{\rule{0ex}{0ex}}=\left(\frac{1}{2}{\right)}^{3}⇒\frac{{r}_{1}}{{r}_{2}}\phantom{\rule{0ex}{0ex}}=\frac{1}{2}$
Now ratio in their surfaces areas $=\frac{4\pi {r}_{1}^{2}}{4\pi {r}_{2}^{2}}\phantom{\rule{0ex}{0ex}}=\frac{{r}_{1}^{2}}{{r}_{2}^{2}}\phantom{\rule{0ex}{0ex}}=\left(\frac{{r}_{1}}{{r}_{2}}{\right)}^{2}\phantom{\rule{0ex}{0ex}}=\frac{1}{2}=\frac{1}{4}$

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