How Can Calculus Help Position the Center of Mass in a Hemiwasher?

[Seda]

Question Details:
I have two circles centered at the origin, one with radius A and the other with radius b.

Looking at the hemiwasher (area between) the circles form above the x axis, find the values of A and B that place the center of mass within the hemiwasher itself, not in the open middle space.

What i think i has solved so far: not necesseraly accurate:
I solved the Y value of the center of mass in terms of A and B to be::Y= (4(A^2+AB+B^2))/(3pi(A+B))

Please Help!

how can I use this to find values of a and b that put the y coordinate of the center of mass between a and b?

B is the smaller radii; the density is constant, so it is irrelvant.

 

[HallsofIvy]

In other words, you want Y between A and B. You need to find A and B such that
$$A\le \frac{4(A^2+AB+ B^2)}{3\pi (A+B)}\le B$$

You won't be able to find specific values of A and B, of course. You want to find a relation between A and B that will guarantee that inequality. I would recommend that you look at
$$A\le \frac{4(A^2+AB+ B^2)}{3\pi (A+B)}$$
and
$$B\ge \frac{4(A^2+AB+ B^2)}{3\pi (A+B)}$$
separately.

Those should give you two relations between A and B. Both need to be satisfied.