= Solution
Treat the gas as a smooth <quasi-one-dimensional flow> through a rigid passage, neglecting axial body forces and heat input. It is the confining medium that is incompressible; the flowing gas must remain compressible. With area $\mathcal A(x)$ and axial speed $u$, continuity and inviscid momentum balance give
$$
\rho u\mathcal A=\dot M,\qquad u\,du=-\frac{dp}{\rho},\qquad dp=c_s^2\,d\rho.
$$
Combining their logarithmic differentials yields the <nozzle area-velocity relation>
$$
\frac{d\mathcal A}{\mathcal A}=(\mathcal M^2-1)\frac{du}{u},\qquad\mathcal M=\frac u{c_s}.
$$
A smooth crossing of $\mathcal M=1$ requires an area extremum. The given radius is positive and has derivative $r'(x)=16r_0x\exp[-16x^2/L^2]/L^2$, which vanishes only at $x=0$ within the slab and changes from negative to positive there. Thus this station is the unique throat, with $r_s=r_0/2$ and $\mathcal A_s=\pi r_0^2/4$. The specified subsonic inlet and supersonic outlet select the smooth accelerating branch, so
$$
\boxed{x_s=0}.
$$
A throat by itself does not guarantee transonic flow for arbitrary reservoir and back <pressures>; that branch is specified here. Before the throat a converging section accelerates <subsonic flow>; beyond it a diverging section accelerates <supersonic flow>.<Nozzle Design by NASA Glenn Research Center>[Supersonic and Subsonic Flow, continuity/momentum derivation of the area-velocity relation and choked throat]{ref}
For the <temperature> task, evaluate the <mass> flux and <sound speed> at the throat using $p=K\rho^\gamma$:
$$
\dot M=\mathcal A_s\rho_sc_s=\mathcal A_s\sqrt{\gamma K}\,\rho_s^{(\gamma+1)/2},\qquad\rho_s=\left(\frac{\dot M}{\mathcal A_s\sqrt{\gamma K}}\right)^{2/(\gamma+1)}.
$$
The ideal-gas relation then gives
$$
\boxed{T_{x_s}=\frac{\mu K}{R^*}\left(\frac{4\dot M}{\pi r_0^2\sqrt{\gamma K}}\right)^{2(\gamma-1)/(\gamma+1)}}.
$$
This expresses the sonic <temperature> in terms of the <entropy> constant, geometry, <mass> flux and composition; no numerical value of $K$ or $\dot M$ is provided. Equivalently $c_s^{\gamma+1}=\gamma K(\dot M/\mathcal A_s)^{\gamma-1}$ and $T_s=\mu c_s^2/(\gamma R^*)$. The slab length drops out because the sonic thermodynamic state depends on the minimum area, not on the axial distance over which a lossless smooth <nozzle> contracts. As a check, conserved stagnation enthalpy gives $T_s=2T_{\rm stag}/(\gamma+1)$.<Mass Flow Choking by NASA Glenn Research Center>[equations 1–10 and 19, sonic mass-flow maximum and temperature/Mach relation]{ref}
The result assumes cross-sectional thermodynamic uniformity and negligible transverse <kinetic energy>. Exact multidimensional flow through a sharply varying passage can have a curved sonic surface; the radius profile alone does not solve that boundary-value problem. The original numerical plot below computes the standard quasi-one-dimensional branch, rather than reproducing the supplied passage sketch.
\Image[/past-exam-of-the-astrophysical-fluid-dynamics-course-of-the-university-of-cambridge/2016/paper-1-transonic-flow.png]
{title=Calculated area, <Mach number> and temperature along the smooth transonic nozzle branch}
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