 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 $v/(a\ell )\le 1/3$, henceEquality 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.