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 ?^@
Answer:
^@ \sqrt[3]{ 2 } : 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. - 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)] ^@ - 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 ^@ - 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 ^@
- ^@ \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} ^@
- Therefore, the required ratio is ^@ \sqrt[3]{ 2 } : 1.^@
