A circle of radius 4 is contained in the plane y=7.

Both circles lie on the surface of a sphere.

What is the radius of the sphere?

The center of the sphere must lie on the line through the two circles at a point equidistant from the circle edges.  Consider the circle formed when a plane halves the sphere along the line.  The problem is now to find the radius of the circle.

The circle has two half chords of 2 and 4, forming right triangles with hypotenuses r, and single legs of 2 and 4.  The other legs must sum to 4, which is the distance separating the chords along the diagonal.  The equation presents itself: 

4=(r^2 - 4)^.5 + (r^2 - 16)^.5

Solving for r:  root(15)/2