Answer:
Among those two medium, light would travel faster in the one with a reflection angle of [tex]32^{\circ}[/tex] (when light enters from the air.)
Explanation:
Let [tex]v_{1}[/tex] denote the speed of light in the first medium. Let [tex]v_{\text{air}}[/tex] denote the speed of light in the air. Assume that the light entered the boundary at an angle of [tex]\theta_{1}[/tex] to the normal and exited with an angle of [tex]\theta_{\text{air}}[/tex]. By Snell's Law, the sine of [tex]\theta_{1}\![/tex] and [tex]\theta_{\text{air}}\![/tex] would be proportional to the speed of light in the corresponding medium. In other words:
[tex]\displaystyle \frac{v_{1}}{v_{\text{air}}} = \frac{\sin(\theta_{1})}{\sin(\theta_{\text{air}})}[/tex].
When light enters a boundary at the critical angle [tex]\theta_{c}[/tex], total internal reflection would happen. It would appear as if the angle of refraction is now [tex]90^{\circ}[/tex]. (in this case, [tex]\theta_{\text{air}} = 90^{\circ}[/tex].)
Substitute this value into the Snell's Law equation:
[tex]\begin{aligned}\frac{v_{1}}{v_{\text{air}}} &= \frac{\sin(\theta_{1})}{\sin(\theta_{\text{air}})} \\ &= \frac{\sin(\theta_{c})}{\sin(90^{\circ})} \\ &= \sin(\theta_{c})\end{aligned}[/tex].
Rearrange to obtain an expression for the speed of light in the first medium:
[tex]v_{1} = v_{\text{air}} \cdot \sin(\theta_{1})[/tex].
The speed of light in a medium (with the speed of light slower than that in the air) would be proportional to the critical angle at the boundary between this medium and the air.
For [tex]0 < \theta < 90^{\circ}[/tex], [tex]\sin(\theta)[/tex] is monotonically increasing with respect to [tex]\theta[/tex]. In other words, for [tex]\!\theta[/tex] in that range, the value of [tex]\sin(\theta)\![/tex] increases as the value of [tex]\theta\![/tex] increases.
Therefore, compared to the medium in this question with [tex]\theta_{c} = 27^{\circ}[/tex], the medium with the larger critical angle [tex]\theta_{c} = 32^{\circ}[/tex] would have a larger [tex]\sin(\theta_{c})[/tex]. such that light would travel faster in that medium.
