A) \[1\,\,:\,\,1\]
B) \[\frac{4\pi }{3}\,\,:\,\,1\]
C) \[{{\left( \frac{\pi }{6} \right)}^{1/3}}\,:\,\,1\,\]
D) \[\frac{1}{2}{{\left( \frac{4\pi }{3} \right)}^{2/3}}\,:\,\,1\,\]
Correct Answer: C
Solution :
\[{{V}_{1}}={{V}_{2}}\] \[\frac{4}{3}\pi {{r}^{3}}\,\,=\,\,{{\ell }^{3}}\] \[r={{\left( \frac{3}{4\pi } \right)}^{1/3}}\,\,\ell \Rightarrow {{r}^{2}}\,\,=\,\,{{\left( \frac{3}{4\pi } \right)}^{2/3}}{{\ell }^{2}}\] \[\frac{dQ}{dt}\,\,\,\propto \,\,A\] \[\Rightarrow \,\,\frac{{{R}_{{{h}_{1}}}}}{{{R}_{{{h}_{2}}}}}\,=\,\frac{4\pi {{r}^{2}}}{6{{t}^{2}}}\] \[=\frac{4\pi }{6}\,{{\left( \frac{3}{4\pi } \right)}^{2/3}}\,\,=\,\,\frac{1}{2}\left( \frac{4\pi }{3} \right){{\,}^{1/3}}:1\]
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