What is the radius of the smallest circle into which will fit four unit half-disks? What arrangement of the half-disks achieves this? How is it proved optimal?
The best arrangement I've found fits in a circle of radius $\frac1{30}(50-5\sqrt{3}+\sqrt{55})$:
Could this actually be optimal? I certainly haven't been able to prove it.
The best lower bound I have is the obvious $\sqrt{2}$ based on comparing areas.
Note on source: I found this question on Quora.
I don't know what the optimal configuration is, but yours isn't. Consider rotating your two-half-disk circle by slightly less than $\pi/2$. You can then slide one of the half-disks so it doesn't touch the outer circle. And then you can adjust the outer circle to make it smaller.
EDIT: I mean that in a configuration like this
you can slide the yellow half-disk.
Oh, and by the way, I believe your large circle has radius $13/8$, not $\frac1{30}(50-5\sqrt{3}+\sqrt{55})$.