Calculating radii (or difference of radii) of concentric spheres with spherical shell volume

Question

I'd like to calculate the radii or the difference of the radii (thickness of spherical shell) of 2 concentric spheres where the only given value is the volume between the concentric spheres (spherical shell). The inner sphere has a constant volume (non-zero).

I know the volume of the spherical shell's equation would be like this:

$V = 4/3\pi(R^3-r^3)$

I thought this formula might be involved for the cubic differences of the radii

$(x^3-y^3) = (x-y)(x^2+xy+y^2)$

Which would lead to this:

$V = 4/3\pi(R-r)(R^2+Rr+r^2)$

I'm not sure where to go from here. I would also be happy to know if there are any alternative methods for the solution.

Answer 1

Your question is underdetermined. For a given volume, we could simply have the outer sphere have that volume and the inner sphere have volume $0$. Or we could move the outer sphere's surface outward and move the inner sphere's surface outward a somewhat greater amount to keep the volume between them constant.

EDIT: For example, suppose the difference in volumes of the two spheres is $\frac{4\pi}{3}\cdot 16^3$. This can be accomplished with radii $0$ and $4$, or with any inner radius you want, say $5$, you can calculate what the outer radius must be:$$\frac{4\pi}{3}R^3-\frac{4\pi}{3}5^3 = \frac{4\pi}{3}\cdot 16^3$$ Which you can solve for $R$. Replace $5$ and $16$ by any values you like and solve for the outer radius needed.