Fit a scalability curve and find where it turns over

A distributed index reports relative capacity $C(p)$ at $p$ shards:

$p$	1	2	4	8	16	32
$C$	1.00	1.96	3.72	6.63	10.08	11.39

1. Fit $C(p) = p/(1+\alpha(p-1)+\beta p(p-1))$ by least squares on $p/C(p) - 1$, which is linear in the two parameters with no intercept.
2. Predict $p^\star$ and the capacity there.
3. The team plans to run 64 shards. What does the model predict, and what would you tell them?