Volume of a sphere section. Is there enough data?

Question

I am thinking about a problem that seems easy but I suspect that is not possible solve it with the available data.

Thanks very much!

Problem is this:

There is a sphere cutted in four equal pieces (see diagram), we know the volume of one and the volume of the half slice a+b (as can see, there are two egual half slice a+b into c). With this data we could know the volume of b?

Answer 1

From the volume of the piece c, you can deduce the radius of the sphere. If the volume of that piece is $V_c$, then $V_c=\frac14\frac43 \pi r^3$.

You can relate the thickness of the shaded region to its volume too by using integration to get the volume of the shaded region. The integral $\frac{\pi}{2}\int_0^t x^2-r^2 dx$ where t is the thickness will do the trick.

Answer 2

Yes, knowing $c$ is enough to get you the radius of the sphere. That gives you the volume of a hemisphere. If you subtract twice the volume $a+b$ you are left with a spherical cap of known volume. From that you can determine the height of the cap, and then the thickness of the slice $a+b$. That gets you $b$.