Diffraction Grating $($a$)[5$pt$]$ In a plane transmission grating, the angle of diffraction for the third$-$order maximum at a wavelength of $532$ nm is $45$ degrees. Determine the number of lines in one centimeter of the grating surface. $($b$)[5$pt$]$ Prove that a grating with $2000$ lines per cm cannot produce a spectrum in the $9$th or higher order for light with a wavelength of $589$ nm$.$