Find an expression, in terms of $r$, for the volume of the largest cube that can possibly fit inside a sphere of radius $r$ cm. Answer ... $\text{cm}^3$

1. Relate the dimensions of the cube to the sphere's radius

Let the side length of the cube be $x$.
The largest cube that fits inside a sphere will have its vertices touching the sphere.
The space diagonal of the cube is equal to the diameter of the sphere, which is $2r$.

2. Express the side length of the cube in terms of $r$

Using Pythagoras' theorem in 3D, the space diagonal $d$ of a cube with side $x$ is given by:
$d^2 = x^2 + x^2 + x^2 = 3x^2$
Since $d = 2r$, we have:
$3x^2 = (2r)^2 = 4r^2$
$x^2 = \frac{4r^2}{3}$
$x = \frac{2r}{\sqrt{3}}$

3. Calculate the volume of the cube

Volume of the cube $V = x^3$
$V = \left(\frac{2r}{\sqrt{3}}\right)^3$
$V = \frac{8r^3}{3\sqrt{3}}$