= Solution
The <Stefan–Boltzmann law> gives $L=4\pi R^2\sigma T_{\rm eff}^4$ for bolometric <luminosity>. Hence
$$
\boxed{\frac{T_s}{T_\ell}=\left[\frac{L_s/L_\ell}{(R_s/R_\ell)^2}\right]^{1/4}.}
$$
With the central values above, this is approximately $1.51$ for the first geometry and $0.67$ for the second. If only one-band <luminosities> are measured, <bolometric corrections> are needed before using this formula for <effective temperatures>.
Two direct methods are fitting the continuum <stellar spectrum> or several photometric colors to a <stellar atmosphere> model, and measuring temperature-sensitive <absorption lines> or line ratios in a <stellar spectrum>. The former includes the approximate <black-body radiation> estimate $\lambda_{\max}T=b$ from <Wien's displacement law>; the latter uses ionization and excitation populations. Observations during total eclipse or spectral disentangling help separate the two <stellar spectra>.
For comparable composition and accessible fuel fractions, the <main-sequence lifetime> is proportional to available nuclear fuel divided by <luminosity>, $t_{\rm MS}\propto M/L$. Estimate the masses from a calibrated <mass-luminosity relation> $L\propto M^\alpha$ in the appropriate mass range. Then
$$
\boxed{\frac{t_s}{t_\ell}\simeq\frac{M_s/M_\ell}{L_s/L_\ell}=\left(\frac{L_s}{L_\ell}\right)^{1/\alpha-1}.}
$$
For illustration $\alpha=3.5$ gives about $1.27$ and $5.6$ for the two graphical <luminosity> ratios. The common <main sequence> approximation is additional physics; eclipse depths alone do not determine masses or ages. A <main sequence> consistency check would also favor the second geometry because ordinary <main-sequence stars> have radius and <effective temperature> increasing with mass.
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