In the image, the yellow and blue horizontal line segments should be ignored.
I only drew them to crop the image to the interesting volumes of the displaced water.
Thus only the blue and yellow curves should be looked at, or considered.

If 0 < r < a, then only the blue curve is valid.
This left part of the blue curve corresponds to spheres which are fully submerged.

If r = a, then the blue and yellow curves are tangent.

If a < r < b, then only the yellow curve is valid.
This right part of the yellow curve corresponds to spheres which fit into the cone, but are only partially submerged.

a = h/(Csc[θ] + 1)
b = h Tan[θ] Sec[θ]

h = 7
θ = π/6