I wish to create 5 identical white circles of some lesser radius, which I will place on top of the black circle and completely obscure (cover) the black circle.

What is the smallest radius which I can make the smaller circles and still meet my requirement?

3.535533906

It seems to me that the side of an inscribed square to the black circle represents the smallest possible diameter of the smaller circle. So, 10 is the diameter of the black circle and also the diagonal of the square. Solving the isoceles right triangle for hypotenuse 10 gives the side as 7.071067812, so the radius is 3.535533906.

Also, just from the diagonal ... 10 times sin45º/2 gives the same answer. So, four white circles around the circumference of the black circle and cover what's left in the middle with the fifth white circle.