= Solution
For nondegenerate <Electrons> in a parabolic <conduction band>,
$$
n(T)=N_c(T)\exp\left[-\frac{\Delta(T)}{k_BT}\right],\qquad \Delta(T)=E_c(T)-\mu(T),\qquad N_c(T)\propto T^{3/2}
$$
when the density-of-states <effective mass> and valley multiplicity are fixed. The <intrinsic carrier extrapolation across a crossover> must therefore specify how the activation <energy> changes with <temperature>.
The usual intended estimate treats the quoted $0.9\,\mathrm{eV}$ as a constant intrinsic activation <energy> between the two temperatures, and identifies the crossover by $n_i(T_c)\simeq n_{\rm ext}$. Then
$$
\frac{n_i(T_c)}{n_i(T_1)}=\left(\frac{T_c}{T_1}\right)^{3/2}\exp\left[-\frac{0.9\,\mathrm{eV}}{k_B}\left(\frac1{T_c}-\frac1{T_1}\right)\right].
$$
For $T_1=350\,\mathrm K$, $T_c=280\,\mathrm K$ and $k_B=8.6173\times10^{-5}\,\mathrm{eV\,K^{-1}}$, the exponent is $-7.46$ and the prefactor is $0.8^{3/2}=0.7155$. Hence
$$
\boxed{n_{\rm ext}\simeq(2.3\times10^{16})(0.7155)e^{-7.46}\simeq9.5\times10^{12}\,\mathrm{cm^{-3}}\quad\text{under the constant-activation approximation}.}
$$
Equivalently, a fixed-gap intrinsic model with $E_g/2=0.9\,\mathrm{eV}$ gives this result. A crossover is gradual: if $n_{\rm ext}$ denotes the net ionized donor density and $n_i(T_c)=n_{\rm ext}$, <charge neutrality> gives the actual <Electron> population $n(T_c)=(1+\sqrt5)n_{\rm ext}/2$, rather than exactly $n_{\rm ext}$.
There are two limits to interpreting the numerical estimate literally. First, the printed data specify $E_c-\mu$ only at $350\,\mathrm K$, without specifying $E_g$ or the valence <density of states>. They do not mathematically determine $\Delta(280\,\mathrm K)$: without the intended activation assumption the answer remains
$$
n_{\rm ext}\simeq n(350)\left(\frac{280}{350}\right)^{3/2}\exp\left[-\frac{\Delta(280)}{k_B(280)}+\frac{0.9\,\mathrm{eV}}{k_B(350)}\right].
$$
Second, using the original PDF's $0.9\,\mathrm{eV}$ and density implies $N_c(350)\simeq2.1\times10^{29}\,\mathrm{cm^{-3}}$, extraordinarily large for a conventional <semiconductor>. This confirms that the numerical data should not be treated as a realistic material determination. No unprinted decimal-point correction has been assumed.
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