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?

Determining Minimum Volume of Cubical Box for Sphere Enclosure

The question asks for the minimum volume of a cubical box required to enclose a sphere with a given radius $r$ cm. To solve this, we need to understand the relationship between the dimensions of the sphere and the smallest possible cube that can contain it.

Understanding Sphere and Cube Dimensions

A sphere is defined by its radius $r$. Its diameter, which is the longest distance across the sphere passing through its center, is $d=2r$.

A cube is a three-dimensional shape with six equal square faces. Its size is determined by its side length, let's call it $s$. The volume of a cube is given by the formula $V_{cube}=s^3$.

For a cubical box to enclose a sphere, the sphere must fit entirely within the boundaries of the cube. The minimum size of the cube occurs when the sphere touches the faces of the cube.

Relating Sphere Diameter to Minimum Cube Side Length

Imagine placing the sphere inside the cube. The sphere will touch the center of each of the cube's six faces. The distance between two opposite faces of the cube is equal to the side length $s$ of the cube. This distance must be at least as large as the diameter $d$ of the sphere for the sphere to fit inside.

Therefore, the minimum side length $s$ of the cubical box must be equal to the diameter of the sphere.

- Sphere Radius = $r$ cm
- Sphere Diameter $d=2\times \text{radius}=2r$ cm
- Minimum Cube Side Length $s=\text{Sphere Diameter}=2r$ cm

Calculating Minimum Cube Volume

Now we can calculate the volume of this minimum-sized cubical box using the formula for the volume of a cube, $V_{cube}=s^3$.

Substitute the minimum side length $s=2r$ into the volume formula:

$V_{cube}=(s{)}^3=(2r{)}^3$

To simplify $(2r{)}^3$, we cube both the coefficient 2 and the variable $r$:

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

Since $2^3=2\times 2\times 2=8$, the volume is:

$V_{cube}=8r^3$

Conclusion

The minimum volume of the cubical box that can enclose a sphere of radius $r$ cm is $8r^3$ cm³.