A sphere of radius 𝑟 cm is packed in a box of cubical shape.

What should be the minimum volume (in cm3 ) of the box that can enclose the sphere?

Sphere in Cubical Box: Minimum Volume Calculation

The problem asks us to find the minimum volume of a cubical box that can completely enclose a sphere with a given radius, r cm.

Understanding the Relationship Between Sphere and Cube

To enclose a sphere perfectly within a cubical box, the dimensions of the box must be such that they accommodate the sphere's widest part, which is its diameter.

- The radius of the sphere is given as r cm.
- The diameter of the sphere is twice its radius. Using LaTeX notation, the diameter is $2r$ cm.
- For the cubical box to enclose the sphere, the length of each side of the cube must be at least equal to the diameter of the sphere.
- Therefore, the minimum side length (let's denote it as s) of the cubical box required is $s=2r$ cm.

Calculating the Minimum Volume of the Box

The volume of a cube is calculated by cubing its side length. The formula for the volume (V) of a cube is $V=s^3$.

Substituting the minimum side length we found:

Minimum Volume $V_{min}=(2r{)}^3$

To calculate this, we cube both the coefficient and the variable:

$V_{min}=2^3\times r^3$

$V_{min}=8\times r^3$

So, the minimum volume of the cubical box is $8r^3$ cm³.

Matching with the Options

Comparing our calculated minimum volume with the given options:

- Option 1: $\frac{r^3}{8}$
- Option 2: $r^3$
- Option 3: $2r^3$
- Option 4: $8r^3$

Our calculated volume, $8r^3$ cm³, matches Option 4.