= Solution
For a dodecyl tail, take $n=12$. The given relations yield $v=0.3502$ nm$^3$ and $\ell=1.672$ nm. The spherical-core condition is $v/(a\ell)\leq1/3$, hence
$$
\boxed{a_{\min}=\frac{3v}{\ell}=0.628\ \mathrm{nm^2}.}
$$
Equality corresponds to a maximally extended tail reaching the core centre. A smaller head-group area would require a spherical radius larger than the available chain length. The estimate assumes all twelve tail carbons contribute to the hydrocarbon core; partial penetration or hydrated head-group geometry would modify it.
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