An optician is performing Young's double$-$slit experiment for his clients. He directs a beam of monochromatic light to a pair of parallel slits, which are separated by $0\mathrm{.}110$ mm from each other. The portion of this light that passes through the slits goes on to form an interference pattern upon a screen, which is $4\mathrm{.}50$ meters distant.

The light is characterized by a wavelength of $577$ nm$.$

$($a$)$

What is the optical path$-$length difference $($in $$m$)$ that corresponds to the fifth$-$order bright fringe on the screen? $($This is the fifth fringe, not counting the central bright band, that one encounters moving from the center out to one side.$)$

$$m

$($b$)$

What path$-$length difference $($in $$m$)$ corresponds to the fifth dark fringe that one encounters when moving out to one side of the central bright fringe?

$$m