A) \[1:{{2}^{\frac{1}{3}}}\]
B) \[{{2}^{\frac{1}{3}}}:1\]
C) \[{{2}^{\frac{1}{3}}}:{{3}^{\frac{1}{3}}}\]
D) \[{{3}^{\frac{1}{2}}}:{{2}^{\frac{1}{2}}}\]
Correct Answer: B
Solution :
Let the height of cylinder = h |
Then, radius of cylinder \[=\frac{h}{2}\] |
Then, radius of cylinder \[=\frac{h}{2}\] |
Also, radius of the sphere, \[r=\frac{h}{2}\] |
\[\therefore \] Volume of cylinder \[=\pi {{\left( \frac{h}{2} \right)}^{2}}h=\frac{\pi {{h}^{3}}}{4}\] |
and volume of sphere material |
\[=\frac{4}{3}\pi {{r}^{3}}=\frac{4}{3}\pi {{\left( \frac{h}{2} \right)}^{3}}=\frac{\pi {{h}^{3}}}{6}\] |
\[\therefore \] Volume of remaining material |
\[=\frac{\pi {{h}^{3}}}{4}-\frac{\pi {{h}^{3}}}{6}=\frac{\pi {{h}^{3}}}{12}\] |
\[\therefore \] Volume of sphere in remaining material, |
\[{{S}_{1}}=\frac{4}{3}\pi {{R}^{3}}\] |
By given condition, \[\frac{4}{3}\pi {{R}^{3}}=\frac{\pi {{h}^{3}}}{12}\Rightarrow {{R}^{3}}=\frac{{{h}^{3}}}{16}\] |
\[\Rightarrow \] \[R=\frac{h}{2\cdot {{2}^{1/3}}}\] |
\[\therefore \] Required ratio |
\[=\frac{r}{R}=\frac{h/2}{h/2\cdot {{2}^{1/3}}}\]\[\Rightarrow \]\[r:R={{2}^{1/3}}:1\] |
You need to login to perform this action.
You will be redirected in
3 sec