when a fish looks up the surface of a perfectly smooth lake ,the surface appears dark except inside a circular area directly above it. calculate the angle that illuminates the region subtends.

Dear Student,
please find below the solution to the asked query:

It will be the angle for which the angle of refraction for a ray coming out of water would be 90 degrees, because if the ray with any higher angle of incidence would be internally reflected (total internal reflection)
$$\begin{gathered} if the angle that the ray makes with vertical be {\theta }_c then \\ {\mathit{n}}_{\mathbf{w}\mathbf{a}\mathbf{t}\mathbf{e}\mathbf{r}}\mathit{s}\mathit{i}\mathit{n}\mathbf{\left(}{\mathit{\theta }}_{\mathbf{c}}\mathbf{\right)}\mathbf{=}{\mathit{n}}_{\mathbf{a}\mathbf{i}\mathbf{r}}\mathit{s}\mathit{i}\mathit{n}\mathbf{\left(}{\mathbf{90}}^{\mathbf{0}}\mathbf{\right)} \\ \mathit{O}\mathit{r}\mathbf{,}\mathbf{ }\mathbf{sin}\mathbf{\left(}{\mathit{\theta }}_{\mathbf{c}}\mathbf{\right)}\mathbf{=}\frac{\mathbf{3}}{\mathbf{4}} \\ {\mathit{\theta }}_{\mathbf{c}}\mathbf{=}\mathbf{48}\mathbf{.}{\mathbf{6}}^{\mathbf{0}} \\ so angle subtended by the illuminated region would be \mathbf{2}\mathbf{\times }\mathbf{48}\mathbf{.}{\mathbf{6}}^{\mathbf{0}}\mathbf{=}\mathbf{97}\mathbf{.}{\mathbf{2}}^{\mathbf{0}} \end{gathered}$$
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