I'm very math illiterate, so please be patient with me if my question sounds silly or misinformed.
I'm trying to find the height and width axis radii of an ellipsoid, when I only know its volume and its length axis radius.
The volume is $15$ mL ($15,000$ mm$^3$) and the length axis radius is $55$ mm.
The calculator on Google gives these formulas to solve for the different variables in an ellipsoid (volume ($V$) and length ($a$), width ($b$), and height ($c$) axis radii).
To solve for the volume:
$$V = \frac{4}{3}\pi abc$$
To solve for the length axis radius ($b$ and $c$ can be swapped with $a$ to solve for those radii):
$$a = 3\frac{V}{4\pi bc}$$
The calculator doesn't return anything unless I provide values for three out of four of these variables.
I've asked for help with this in a few other places to cast a wide net, but the only person to respond so far says that s/he doesn't think it's possible to solve for $b$ and $c$ individually. I don't want to give up hope just yet, though, so I'm asking here also.
I have the feeling that I should at least be able to find a range of possible values that $b$ and $c$ would need to be in order to produce an ellipsoid that has the given $V$ and $a$.
Is it possible to do this, and if so, how?
A related and embarrassing followup question:
The value of the variable $V$ is in mL/mm$^3$ and the variables $a$, $b$, and $c$ are in mm. Should I be entering $V = 15,000$ (or $15$) and $a = 55$ in the calculator for the values of those variables, or should I be converting these values to different units?
I don't even know what tags to use for this question, so I only picked conic-sections.