$($a$)$ A light ray is incident on the left side of the prism shown in the figure below. The prism is a right triangle, with two of its angles measuring $\backslash (45^\{\backslash$circ$\}\backslash ).$ The transmitted ray that exits the hypotenuse of the prism makes an angle of $\backslash (\backslash$theta$=21\mathrm{.}7^\{\backslash$circ$\}\backslash )$ with the direction of the incident beam. What is the index of refraction of the prism? $($Round your answer to at least two decimal places.$)$

$(1)$

What is the angle of incidence of the light at the hypotenuse of the prism? What is the angle of refraction at the hypotenuse, and how does it relate to the given angle? How does Snell's law relate the incident and refractive angles?

$($b$)$ What If$?$ In part $($a$),$ we assumed the ray was monochromatic. Consider instead the case where the ray was composed of white light. Because the index of refraction differs for different wavelengths, the white light would be dispersed into constituent colors. Assume the index of refraction for blue wavelengths is $\backslash (1\mathrm{.}01$ n $\backslash )$ and for red wavelengths it is $\backslash (0\mathrm{.}99$ n $\backslash ),$ where $\backslash ($ n $\backslash )$ is the index of refraction found in part $($a$).$ What is the angular spread $($in degrees$)$ between red and blue light exiting the prism?

Using the index of refraction found in part $($a$),$ can you apply Snell's law to find the refraction angle of the red light? Of the blue light? What is the angle between them? $\backslash (\{\}^\{\backslash$circ$\}\backslash )$