For optimal operation of the refractometer, the light beam needs to strike the surface between the prism and the sample at a certain angle range $($i$.$e$.,$ the angle between rays $1$ and $3),$ which can be controlled with an aperture in front of the focusing lens. Assume that the light source is a point source emitting light within an angle range that fills the entire diameter of the lens with a fully open aperture $($i$.$e$.,$ aperture diameter $=$ lens diameter$).$ What is the angle range of the focused beam of rays after the lens that hits the surface between the prism and the sample $($see figure$)$ for the following data:

Lens diameter: $20$ mm

Distance between the light source and the lens: $75$ mm

Focal length of the lens: $30$ mm

Provide an answer with both a calculation and a ray construction illustrating how the rays travel from the light source, through the lens, and to the surface between the prism and the sample. You can assume that the prism is a hemispherical shape with a curved surface facing the lens, so that all rays from the lens hit the prism normally