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 ... cm^3

1. Relate the cube's diagonal to the sphere's diameter

For the largest cube, the longest diagonal of the cube just fits across the sphere.

So the space diagonal of the cube is the diameter of the sphere:

$d=2r.$

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

Let the side length of the cube be $x$ cm.

For a cube, its space diagonal is

$d=\sqrt{x^2+x^2+x^2}=x\sqrt{3}.$

Hence

$x\sqrt{3}=2r$

$x=\frac{2r}{\sqrt{3}}.$

3. Find the volume of the cube

Volume of cube:

$V=x^3$

$V=\left(\frac{2r}{\sqrt{3}}\right)^3$

$V=\frac{8r^3}{3\sqrt{3}}.$