A) \[{{2}^{1/3}}:1\]
B) \[{{2}^{-3/2}}:1\]
C) \[{{2}^{2/3}}:1\]
D) \[{{4}^{-2/3}}:1\]
Correct Answer: A
Solution :
Volume of sphere\[=\frac{4}{3}\pi {{r}_{1}}^{3}\] Volume of hemisphere \[=\frac{2}{3}\pi {{r}_{2}}^{3}\] Now, the volume of both are equal\[\frac{4}{3}\pi {{r}_{1}}^{3}=\frac{2}{3}\pi {{r}_{2}}^{3}\] \[3\sqrt{2}\,\,{{r}_{1}}={{r}_{2}}\] ?(i) Now, curved surface area of sphere\[=4\pi {{r}_{1}}^{2}\] curved surface area of hemisphere \[=2\pi {{r}_{2}}^{2}\] \[\therefore \]Their ratio \[=\frac{\text{Curved}\,\,\text{surface}\,\,\text{area}\,\,\text{of}\,\,\text{sphere}}{\text{Curved}\,\,\text{surface}\,\,\text{area}\,\,\text{of}\,\text{hemisphere}}\] \[=\frac{4\pi {{r}_{1}}^{2}}{2\pi {{r}_{2}}^{2}}=\frac{4\pi {{r}_{1}}^{2}}{2\pi {{2}^{2/3}}{{r}_{1}}^{2}}\] \[=\frac{2}{{{2}^{2/3}}}=\frac{2\times {{2}^{-2/3}}}{1}=\frac{{{2}^{\left( 1-\frac{2}{3} \right)}}}{1}={{2}^{1/3}}:1\]
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