How do they arrive at the conclusion that you should maximize r_a first and then see how large r_b can be to fill up the rest? It makes sense as a good guess, but are you supposed to be able prove it mathematically aswell?
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No, you aren’t supposed to prove it mathematically. However to see why this is true, observe that r_a + r_b = \text{const}, while volume is \propto r_a^3 + r_b^3. Because of the cubic power, maximal ratio gives the maximal filling fraction.
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