To find the dimensions of the cylinder with the largest volume that can fit inside the given cone, we need to maximize the volume of the cylinder.

Let's denote the radius of the cylinder as r and the height of the cylinder as h. Since the axis of the cylinder lies on the axis of the cone, the height of the cylinder will be equal to the height of the cone (h = 5).

To find the optimal radius of the cylinder, we can consider the similar triangles formed by the cone and the cylinder. The ratio of the radius of the cone to the radius of the cylinder should be equal to the ratio of the height of the cone to the height of the cylinder. In this case, the ratio is:

3 / r = 5 / h

Substituting the known values, we have:

3 / r = 5 / 5
3 / r = 1

Solving for r, we find:

r = 3

Therefore, the radius of the cylinder is 3 units.

The height of the cylinder is equal to the height of the cone, which is 5 units.

Hence, the dimensions of the cylinder of the largest volume that can fit inside the cone are:
Radius = 3 units
Height = 5 units.