In a Young's double$-$slit experiment, two parallel slits with a slit separation of $0\mathrm{.}135$ mm are illuminated by light of wavelength $574$ nm$,$ and the interference pattern is observed on a screen located $4\mathrm{.}45$ m from the slits.

$($a$)$ What is the difference in path lengths from each of the slits to the location of the center of a third$-$order bright fringe on the screen?

Your response differs significantly from the correct answer. Rework your solution from the beginning and check each step carefully. $$m

$($b$)$ What is the difference in path lengths from the two slits to the location of the center of the third dark fringe away from the center of the pattern?

Your response differs significantly from the correct answer. Rework your solution from the beginning and check each step carefully. $$m