A ray in air is incident on the horizontal top face of a rectangular glass block of refractive index √(3) at the angle for which the reflected light…

Polarisation by reflection occurs at Brewster's angle $\theta_B$ with $\tan\theta_B=n=\sqrt3$, so $\theta_B=60^\circ$. Snell's law gives $\sin\theta_r=\sin60^\circ/\sqrt3=1/2$, so the refracted ray makes $30^\circ$ with the vertical normal of the top face. The side face is vertical, so its normal is horizontal and the ray meets it at an incidence angle $90^\circ-30^\circ=60^\circ$ (equivalently, $\theta_i=\theta_B$, since $\theta_r=90^\circ-\theta_B$). Total internal reflection needs $\sin60^\circ \ge n_\ell/\sqrt3$, i.e. $n_\ell\le \sqrt3\cdot\frac{\sqrt3}{2}=1.50$. The largest allowed value is $1.50$. The value $0.87$ comes from using $30^\circ$ as the incidence angle, $1.22$ from a wrong square root, and $1.73$ would mean no index contrast at all, so there would be no total reflection.