= Solution
For a <symmetric Flory-Huggins polymer blend>, the mixing <free energy> per lattice site is
$$
\frac{f}{k_BT}=\frac{\phi\ln\phi+(1-\phi)\ln(1-\phi)}N+\chi\phi(1-\phi).
$$
The critical conditions $f''=0$ and $d^3f/d\phi^3=0$ give $\phi_c=1/2$ and $\chi_c=2/N=0.002$. Therefore
$$
\boxed{T_c=\frac{48.2}{0.032+0.002}=1.418\times10^3\,\mathrm K.}
$$
Since $d\chi/dT<0$, heating improves miscibility: this is an \b[<upper critical solution temperature>] in the stipulated model. The very high value is an extrapolation of the supplied interaction-parameter fit, not evidence that an intact experimental blend can be brought reversibly to this <temperature>; chemical stability and the fitted <temperature> range must be checked before interpreting it as an accessible transition.
At $293.15\,\mathrm K$, $\chi=0.132421$ and $N\chi\simeq132$, well into strong segregation. For the <polymer-blend interface width>, state the convention explicitly. In the equal-segment, long-chain square-gradient approximation,
$$
\frac{F}{k_BT\rho_0}=\int dz\left[\chi\phi(1-\phi)+\frac{a^2(\phi')^2}{24\phi(1-\phi)}\right].
$$
Its first integral gives $\phi'=\sqrt{24\chi}\,\phi(1-\phi)/a$, so
$$
\phi(z)=\frac{1+\tanh(z/\ell)}2,\qquad
\ell=\frac{a}{\sqrt{6\chi}}=0.752\,\mathrm{nm}.
$$
The inverse-central-slope width is consequently
$$
\boxed{w\equiv\frac1{\phi'(0)}=2\ell=1.50\,\mathrm{nm}.}
$$
A convention calling the hyperbolic-tangent scale itself the width instead quotes $0.752\,\mathrm{nm}$; these describe the same profile. The predicted interface spans only a few statistical segments, so the <continuum approximation> for gradients is quantitatively marginal. Intrinsic composition width and experimentally broadened width also differ because roughness and finite spatial resolution can broaden a measured profile.
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