I actually reproduced your answer, but I had to cut a hole in the middle of the water. No good. I think Moses was the last one known to have pulled that trick.
I suggest you rethink your logic and try a different equation. It's a hemisphere and you are filling to 1/2. How can you get more than half the radius? Are you filling just the top part? That could be a gravity problem.
Think about what you are saying. "Smaller hemisphere"? What is that? "Hemisphere" means something. Don't try to make it mean something else.
There is a reason why you studied those volumes of solids of revolution. Here's your chance. You can do it the REALLY hard way, if you studied only shells:
to get the appropriate x-value. The associated value of 'y' is what you need.
Or, you can solve for it more directly if you are adept at the y-axis.
Note: I'm not quite convinced on part 'a'. How does the water get to the top layer? I think tesseracting is out of the question. Are you sure it doesn't have to lift whatever water is already in there? Maybe it mixes instantaneously, or some other dubious assumption. It's an okay practice problem, I suppose, but I think I would not loan you my tractor to provide the hydraulics for this assignment.