A student is performing Young's double$-$slit experiment for his mentor. He directs a beam of single$-$wavelength light to a pair of parallel slits, which are separated by $0\mathrm{.}130$ 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 $580$ nm$.$

$($a$)$

What is the optical path$-$length difference $($in $$m$)$ that corresponds to the fourth$-$order bright fringe on the screen? $($This is the fourth 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 fourth dark fringe that one encounters when moving out to one side of the central bright fringe?

$$m