### The diameter of a solid metallic right circular cylinder is equal to its height. After cutting out the largest possible solid sphere $S$ from the cylinder, the remaining material is recast to form a solid sphere $S_1$. What is the ratio of the radius of the sphere $S$ to that of sphere $S_1 ?$

$\sqrt[3]{ 2 } : 1$

Step by Step Explanation:
1. Given, the diameter of the cylinder $=$ height of the cylinder
\begin{align} & i.e. h = 2r && ...(1) \end{align}
We need to know the formula for calculating the volume of a cylinder and the volume of a sphere for this question.
2. The volume of a cylinder with radius $r$ and height \begin{align} h &= \pi r^2h \end{align}
Volume of the given cylinder $= 2\pi r^3 \space \space \space \space \space [\text{ Using equation } (1)]$
3. The radius sphere \begin{align} & S = r && [ \text{ Because } h = 2r ] \end{align}
The volume of the sphere $S = \dfrac{ 4 }{ 3 } \pi r^3$
Therefore, the volume of the remaining material $= 2 \pi r^3 - \dfrac{ 4 }{ 3 } \pi r^3 = \dfrac{ 2 }{ 3 } \pi r^3$
4. The remaining material is recast to form a solid sphere $S_1.$
Let the radius of $S_1 = r_1$
The volume of $S_1 = \dfrac{ 2 }{ 3 } \pi r^3$
5. \begin{align} & \dfrac{ \text{ Radius of the sphere } S_1 }{ \text{ Radius of the sphere } S_2 } = \sqrt[3]{ \dfrac{ \text{ Volume of the sphere } S_1 }{ \text{ Volume of the sphere } S_2 } } \\ \implies & \dfrac{ r }{ r_1 } = \sqrt[3]{ \dfrac{ \dfrac{ 4 }{ 3 } \pi r^3 }{ \dfrac{ 2 }{ 3 } \pi r^3 } } \\ \implies & \dfrac{ r }{ r_1 } = \sqrt[3]{ \dfrac{ 2 }{ 1 } } \\ \end{align}
6. Therefore, the required ratio is $\sqrt[3]{ 2 } : 1.$