An empty X$-$band $(8\mathrm{.}212\mathrm{.}4$ GHz$)$ rectangular waveguide, with dimensions of $2\mathrm{.}286$ cm by $1\mathrm{.}016$ cm$,$ is to be connected to an X$-$band waveguide of the same dimensions but filled with lossless polystyrene $($r $=2\mathrm{.}56).$ To avoid reflections, an X$-$band waveguide $($of the same dimensions$)$ quarter$-$wavelength long section $($Recall EE$382)$ is inserted between the two. Assume dominant$-$mode propagation and that matching is to be made at $10$ GHz$.$ Determine the: $($a$)$ Wave impedance of the quarter$-$wavelength section waveguide. $($b$)$ Dielectric constant of the lossless medium that must be used to fill the quarterwavelength section waveguide. $($c$)$ Length $($in cm$)$ of the quarter$-$wavelength