question_answer

The diameter of a solid metallic right circular cylinder is equal to its height. After cutting out the largest possible solid sphere S from this cylinder the remaining material is recast to form a solid sphere\[{{S}_{1}}.\] What is the ratio of the radius of sphere S to that of sphere \[{{S}_{1}}?\] 

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

\[=\frac{r}{R}=\frac{h/2}{h/2\cdot {{2}^{1/3}}}\]\[\Rightarrow \]\[r:R={{2}^{1/3}}:1\]