| A large solid metallic cylinder whose radius and height are equal to each other is to be melted and 48 identical solid balls are to be recast from the liquid metal, so formed. What is the ratio of the radius of a ball to the radius of the cylinder? |
A) 1 : 16
B) 1 : 12
C) 1 : 8
D) 1 : 4
Correct Answer: D
Solution :
| Volume of cylinder \[=\pi r_{1}^{2}h\] |
| Volume of sphere \[=\frac{4}{3}\pi r_{2}^{3}\] |
| Number of spheres \[=48\] |
| \[\therefore \]\[\frac{\text{Volume}\,\,\text{of}\,\,\text{cylinder}}{\text{Volume}\,\,\text{of}\,\,\text{sphere}}=\frac{\pi r_{1}^{2}h}{\frac{4}{3}\pi r_{2}^{3}}\] |
| \[\Rightarrow \]\[\frac{\pi r_{1}^{2}h}{\frac{4}{3}\pi r_{2}^{3}}=48\]\[\Rightarrow \]\[\frac{\pi r_{1}^{3}}{\frac{4}{3}\pi r_{2}^{3}}=48\] \[[\because {{r}_{1}}=h]\] |
| \[\Rightarrow \]\[\frac{3}{4}{{\left( \frac{{{r}_{1}}}{{{r}_{2}}} \right)}^{3}}=48\]\[\Rightarrow \]\[{{\left( \frac{{{r}_{1}}}{{{r}_{2}}} \right)}^{3}}=\frac{48\times 4}{3}\] |
| \[\Rightarrow \]\[{{\left( \frac{{{r}_{1}}}{{{r}_{2}}} \right)}^{3}}=64\]\[\Rightarrow \]\[\frac{{{r}_{1}}}{{{r}_{2}}}=4\]\[\Rightarrow \]\[\frac{{{r}_{2}}}{{{r}_{1}}}=\frac{1}{4}\] |
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