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?

Take a cross-section of the sphere right through the middle, and containing the y-axis.  The result is a circle of equal radius to the sphere.

The circle includes points (4,7), (-4,7), (2,3), and (-2,3).

There was a point called the circumcenter(?) or something like that, and this point is the center of our circle.  To get this, find where all the perpendicular bisectors of the segments intersect.

Math work:
midpoint= ((4+2)/2,(7+3)/2)=(3,5)
slope= (7-3)/(4-2)=4/2=2
perpendicular=-1/2
equation= y=-x/2+13/2

Since the center is obviously on the y-axis, and the equation intersects at (0,6.5), this is the center.  It is sqrt(65)/2 away from all the points, so this is the radius we are looking for!