= Solution
The calculated <rotational constant> gives $\theta_{\rm rot}=1.18721\ \mathrm K$. At room <temperature>, taken as $300\ \mathrm K$, the continuous maximum is $J_*=10.7404$; comparing the adjacent integer populations gives \b[$J_{\max}=11$]. Taking $298.15\ \mathrm K$ instead gives $J_*=10.7057$ and the same integer answer.
At $8.7\ \mathrm K$, $J_*=1.41417$. In particular,
$$
\frac{N_2}{N_1}=\frac53e^{-4\theta_{\rm rot}/T}=0.966\ldots<1,
$$
and $N_1>N_0$, so \b[$J_{\max}=1$]. The integer comparison avoids treating a noninteger <angular momentum quantum number> as a physical state.
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