The plane cut equal areas if it cuts the two solids in circles of the same radius. That allows us to make this a plane geometry problem by erecting a vertical plane containing the axis of the cone and the center of the circle. To simplify the equation of the resulting circle (on the surface of the sphere), let's put the origin at the center of the circle/sphere. The top and bottom of the sphere are  then each 10 cm away from this midline while the apex of the cone is on the midline.

The equation of the circle is x^2 + y^2 = 100 and in this instance x represents the radius of the circle the sphere cuts off on the horizontal cutting plane mentioned in the puzzle.

The cone (triangle cross section that is) is entirely below the x-axis except for the apex, which is on the x-axis. Half the distance between the two sides with the same value of y constitutes the radius of the cross-section circle of the cone, and that is exactly -y/2.

So we need to solve x^2 + y^2 = 100 where x = -y/2 and x is positive:

1.25*y^2 = 100

y^2 = 80

y = -sqrt(80)

Since the bottom is 10 cm down from the origin, the cutting plane is 10 - sqrt(80) ~= 1.05572809000084 cm up from the floor.