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.
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.