Spheres

Geometry Level 2

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 $Q.$

What is the ratio of the radius of sphere $S$ to that of sphere $Q?$

1 :

2^{\frac{1}{3}}

2^{\frac{1}{3}}

3^{\frac{1}{3}}

2^{\frac{1}{3}}

: 1

3^{\frac{1}{3}}

2^{\frac{1}{3}}

1 solution

A Former Brilliant Member
Apr 1, 2018

$\text{volume of the sphere} = \dfrac{4}{3}\pi r^3$

$\text{volume of the cylinder} = \pi r^2 (2r) = 2\pi r^3$

$\text{volume of remaining material} = 2\pi r^3 - \dfrac{4}{3}\pi r^3 = \dfrac{2}{3} \pi r^3$

Let $R$ be the radius after the remaining material is cast into sphere.

$\dfrac{4}{3}\pi R^3 = \dfrac{2}{3}\pi r^3$

$\dfrac{r^3}{R^3} = \dfrac{4}{3} \times \dfrac{3}{2} = 2$

$\boxed{\dfrac{r}{R}=\dfrac{2^{\frac{1}{3}}}{1}}$

Spheres

1 solution

0 pending reports